# Find if a point is within a triangle

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Find if a point is within a triangle
You are encouraged to solve this task according to the task description, using any language you may know.

Find if a point is within a triangle.

Task
•   Assume points are on a plane defined by (x, y) real number coordinates.
•   Given a point P(x, y) and a triangle formed by points A, B, and C, determine if P is within triangle ABC.
•   You may use any algorithm.
•   Bonus: explain why the algorithm you chose works.

Related tasks

Also see
• Discussion of several methods. [[1]]
• Determine if a point is in a polygon [[2]]
• Triangle based coordinate systems [[3]]
• Wolfram entry [[4]]

## 11l

Translation of: Kotlin
```V EPS = 0.001
V EPS_SQUARE = EPS * EPS

F side(p1, p2, p)
R (p2.y - p1.y) * (p.x - p1.x) + (-p2.x + p1.x) * (p.y - p1.y)

F distanceSquarePointToSegment(p1, p2, p)
V p1P2SquareLength = sqlen(p2 - p1)
V dotProduct = dot(p - p1, p2 - p1) / p1P2SquareLength
I dotProduct < 0
R sqlen(p - p1)
I dotProduct <= 1
V pP1SquareLength = sqlen(p1 - p)
R pP1SquareLength - dotProduct * dotProduct * p1P2SquareLength
R sqlen(p - p2)

T Triangle
(Float, Float) p1, p2, p3

F (p1, p2, p3)
.p1 = p1
.p2 = p2
.p3 = p3

F String()
R ‘Triangle[’(.p1)‘, ’(.p2)‘, ’(.p3)‘]’

F.const pointInTriangleBoundingBox(p)
V xMin = min(.p1.x, min(.p2.x, .p3.x)) - :EPS
V xMax = max(.p1.x, max(.p2.x, .p3.x)) + :EPS
V yMin = min(.p1.y, min(.p2.y, .p3.y)) - :EPS
V yMax = max(.p1.y, max(.p2.y, .p3.y)) + :EPS
R !(p.x < xMin | xMax < p.x | p.y < yMin | yMax < p.y)

F.const nativePointInTriangle(p)
V checkSide1 = side(.p1, .p2, p) >= 0
V checkSide2 = side(.p2, .p3, p) >= 0
V checkSide3 = side(.p3, .p1, p) >= 0
R checkSide1 & checkSide2 & checkSide3

F.const accuratePointInTriangle(p)
I !.pointInTriangleBoundingBox(p)
R 0B
I .nativePointInTriangle(p)
R 1B
I distanceSquarePointToSegment(.p1, .p2, p) <= :EPS_SQUARE
R 1B
I distanceSquarePointToSegment(.p2, .p3, p) <= :EPS_SQUARE
R 1B
R distanceSquarePointToSegment(.p3, .p1, p) <= :EPS_SQUARE

F test(t, p)
print(t)
print(‘Point ’p‘ is within triangle ? ’(I t.accuratePointInTriangle(p) {‘true’} E ‘false’))

V p1 = (1.5, 2.4)
V p2 = (5.1, -3.1)
V p3 = (-3.8, 1.2)
V tri = Triangle(p1, p2, p3)
test(tri, (0.0, 0.0))
test(tri, (0.0, 1.0))
test(tri, (3.0, 1.0))
print()
p1 = (1.0 / 10, 1.0 / 9)
p2 = (100.0 / 8, 100.0 / 3)
p3 = (100.0 / 4, 100.0 / 9)
tri = Triangle(p1, p2, p3)
V pt = (p1.x + 3.0 / 7 * (p2.x - p1.x), p1.y + 3.0 / 7 * (p2.y - p1.y))
test(tri, pt)
print()
p3 = (-100.0 / 8, 100.0 / 6)
tri = Triangle(p1, p2, p3)
test(tri, pt)```
Output:
```Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 0) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 1) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3, 1) is within triangle ? false

Triangle[(0.1, 0.111111111), (12.5, 33.333333333), (25, 11.111111111)]
Point (5.414285714, 14.349206349) is within triangle ? true

Triangle[(0.1, 0.111111111), (12.5, 33.333333333), (-12.5, 16.666666667)]
Point (5.414285714, 14.349206349) is within triangle ? true
```

## Ada

This uses a determinant method to calculate the area of triangles, and tests whether or not a point is in a triangle by adding up the areas of the triangles formed by each side of the triangle with the point in question, and seeing if the sum matches the whole.

It uses a generic two-dimensional geometry, that could be affine, euclidean, or a lot stranger than that. You only need to specify the type of one dimension, and the library should handle the rest. Edge cases probably exist where they shouldn't, as the area formula might add some imprecision.

```-- triangle.ads
generic
type Dimension is private;
Zero, Two: Dimension;
with function "*"(Left, Right: in Dimension) return Dimension is <>;
with function "/"(Left, Right: in Dimension) return Dimension is <>;
with function "+"(Left, Right: in Dimension) return Dimension is <>;
with function "-"(Left, Right: in Dimension) return Dimension is <>;
with function ">"(Left, Right: in Dimension) return Boolean is <>;
with function "="(Left, Right: in Dimension) return Boolean is <>;
with function Image(D: in Dimension) return String is <>;
package Triangle is

type Point is record
X: Dimension;
Y: Dimension;
end record;

type Triangle_T is record
A,B,C: Point;
end record;

function Area(T: in Triangle_T) return Dimension;

function IsPointInTriangle(P: Point; T: Triangle_T) return Boolean;

function Image(P: Point) return String is
("(X="&Image(P.X)&", Y="&Image(P.Y)&")")
with Inline;

function Image(T: Triangle_T) return String is
("(A="&Image(T.A)&", B="&Image(T.B)&", C="&Image(T.C)&")")
with Inline;
end;
```
```-- triangle.adb

package body Triangle is
function Area(T: in Triangle_T) return Dimension
is
tmp: Dimension;
begin
tmp:=((T.B.X*T.C.Y-T.C.X*T.B.Y)-(T.A.X*T.C.Y-T.C.X*T.A.Y)+(T.A.X*T.B.Y-T.B.X*T.A.Y))/Two;
if tmp>Zero then
return tmp;
else
return Zero-tmp;
end if;
end Area;

function IsPointInTriangle(P: Point; T: Triangle_T) return Boolean
is
begin
return Area(T)=Area((T.A,T.B,P))+Area((T.A,P,T.C))+Area((P,T.B,T.C));
end IsPointInTriangle;
end;
```
```-- test_triangle.adb
with Ada.Text_IO;
use Ada.Text_IO;
with Triangle;

procedure test_triangle
is
package affine_tri is new Triangle(Dimension=>Integer, Zero=>0,Two=>2, Image=>Integer'Image);
use affine_tri;
tri1: Triangle_T:=((1,0),(2,0),(0,2));
tri2: Triangle_T:=((-1,0),(-1,-1),(2,2));
origin: Point:=(0,0);
begin
Put_Line("IsPointInTriangle("&Image(origin)&", "&Image(tri1)&") yields "&IsPointInTriangle(origin,tri1)'Image);
Put_Line("IsPointInTriangle("&Image(origin)&", "&Image(tri2)&") yields "&IsPointInTriangle(origin,tri2)'Image);
end test_triangle;
```
Output:
```IsPointInTriangle((X= 0, Y= 0), (A=(X= 1, Y= 0), B=(X= 2, Y= 0), C=(X= 0, Y= 2))) yields FALSE
IsPointInTriangle((X= 0, Y= 0), (A=(X=-1, Y= 0), B=(X=-1, Y=-1), C=(X= 2, Y= 2))) yields TRUE
```

## ALGOL 68

Started out as
Translation of: Common Lisp

With additional material

Translation of: Wren
```BEGIN # determine whether a point is within a triangle or not #
# tolerance for the accurate test #
REAL eps         = 0.001;
REAL eps squared = eps * eps;
# mode to hold a point #
MODE POINT = STRUCT( REAL x, y );
# returns a readable representation of p #
OP    TOSTRING = ( POINT p )STRING: "[" + fixed( x OF p, -8, 4 ) + "," + fixed( y OF p, -8, 4 ) + "]";
# returns 1 if p is to the right of the line ( a, b ), -1 if it is to the left and 0 if it is on it #
PROC side of line = ( POINT p, a, b )INT:
SIGN ( ( ( x OF b - x OF a ) * ( y OF p - y OF a ) )
- ( ( y OF b - y OF a ) * ( x OF p - x OF a ) )
);
# returns the minimum of a and b #
PROC min = ( REAL a, b )REAL: IF a < b THEN a ELSE b FI;
# returns the maximum of a and b #
PROC max = ( REAL a, b )REAL: IF a > b THEN a ELSE b FI;
# returns TRUE if p is within the bounding box of the triangle a, b, c, FALSE otherwise #
PROC point inside bounding box of triangle = ( POINT p, a, b, c )BOOL:
BEGIN
REAL x min = min( x OF a, min( x OF b, x OF c ) );
REAL y min = min( y OF a, min( y OF b, y OF c ) );
REAL x max = max( x OF a, max( x OF b, x OF c ) );
REAL y max = max( y OF a, max( y OF b, y OF c ) );
x min <= x OF p AND x OF p <= x max AND y min <= y OF p AND y OF p <= y max
END # point inside bounding box of triangle # ;
# returns the squared distance between p and the line a, b #
PROC distance square point to segment = ( POINT p, a, b )REAL:
IF REAL a b square length = ( ( x OF b - x OF a ) ^ 2 ) + ( ( y OF b - y OF a ) ^ 2 );
REAL dot product = ( ( ( x OF p - x OF a ) ^ 2 ) + ( ( y OF p - y OF a ) ^ 2 ) ) / a b square length;
dot product < 0
THEN ( ( x OF p - x OF a ) ^ 2 ) + ( ( y OF p - y OF a ) ^ 2 )
ELIF dot product <= 1
THEN ( ( x OF a - x OF p ) ^ 2 ) + ( ( y OF a - y OF p ) ^ 2 )
- ( dot product * dot product * a b square length )
ELSE ( ( x OF p - x OF b ) ^ 2 ) + ( ( y OF p - y OF b ) ^ 2 )
FI # distance square point to segment # ;
# returns TRUE if p is within the triangle defined by a, b and c, FALSE otherwise #
PROC point inside triangle = ( POINT p, a, b, c )BOOL:
IF NOT point inside bounding box of triangle( p, a, b, c )
THEN FALSE
ELIF INT side of ab = side of line( p, a, b );
INT side of bc = side of line( p, b, c );
side of ab /= side of bc
THEN FALSE
ELIF side of ab = side of line( p, c, a )
THEN TRUE
ELIF distance square point to segment( p, a, b ) <= eps squared
THEN TRUE
ELIF distance square point to segment( p, b, c ) <= eps squared
THEN TRUE
ELSE distance square point to segment( p, c, a ) <= eps squared
FI # point inside triangle # ;
# test the point inside triangle procedure #
PROC test point = ( POINT p, a, b, c )VOID:
print( ( TOSTRING p, " in ( ", TOSTRING a, ", ", TOSTRING b, ", ", TOSTRING c, ") -> "
, IF point inside triangle( p, a, b, c ) THEN "true" ELSE "false" FI
, newline
)
);
# test cases as in Commpn Lisp #
test point( ( 0, 0 ), ( 1.5, 2.4 ), ( 5.1, -3.1 ), ( -3.8, 1.2 ) );
test point( ( 0, 1 ), ( 1.5, 2.4 ), ( 5.1, -3.1 ), ( -3.8, 1.2 ) );
test point( ( 3, 1 ), ( 1.5, 2.4 ), ( 5.1, -3.1 ), ( -3.8, 1.2 ) );
test point( ( 5.414286, 14.349206 ), ( 0.1, 0.111111 ), ( 12.5, 33.333333 ), (  25.0, 11.111111 ) );
test point( ( 5.414286, 14.349206 ), ( 0.1, 0.111111 ), ( 12.5, 33.333333 ), ( -12.5, 16.666667 ) );
# additional Wren test cases #
test point( ( 5.4142857142857, 14.349206349206 )
, ( 0.1, 0.11111111111111 ), ( 12.5, 33.333333333333 ), ( 25, 11.111111111111 )
);
test point( ( 5.4142857142857, 14.349206349206 )
, ( 0.1, 0.11111111111111 ), ( 12.5, 33.333333333333 ), ( -12.5, 16.666666666667 )
)
END```
Output:
```[  0.0000,  0.0000] in ( [  1.5000,  2.4000], [  5.1000, -3.1000], [ -3.8000,  1.2000]) -> true
[  0.0000,  1.0000] in ( [  1.5000,  2.4000], [  5.1000, -3.1000], [ -3.8000,  1.2000]) -> true
[  3.0000,  1.0000] in ( [  1.5000,  2.4000], [  5.1000, -3.1000], [ -3.8000,  1.2000]) -> false
[  5.4143, 14.3492] in ( [  0.1000,  0.1111], [ 12.5000, 33.3333], [ 25.0000, 11.1111]) -> true
[  5.4143, 14.3492] in ( [  0.1000,  0.1111], [ 12.5000, 33.3333], [-12.5000, 16.6667]) -> false
[  5.4143, 14.3492] in ( [  0.1000,  0.1111], [ 12.5000, 33.3333], [ 25.0000, 11.1111]) -> true
[  5.4143, 14.3492] in ( [  0.1000,  0.1111], [ 12.5000, 33.3333], [-12.5000, 16.6667]) -> false
```

## AutoHotkey

```T := [[1.5, 2.4], [5.1, -3.1], [-3.8, 1.2]]
for i, p in [[0, 0], [0, 1], [3, 1], [5.4142857, 14.349206]]
result .= "[" p.1 ", " p.2 "]  is within triangle?`t" (TriHasP(T, p) ? "ture" : "false") "`n"
MsgBox % result
return

TriHasP(T, P){
Ax := TriArea(T.1.1, T.1.2,  T.2.1, T.2.2,  T.3.1, T.3.2)
A1 := TriArea(P.1  , P.2  ,  T.2.1, T.2.2,  T.3.1, T.3.2)
A2 := TriArea(T.1.1, T.1.2,  P.1  , P.2  ,  T.3.1, T.3.2)
A3 := TriArea(T.1.1, T.1.2,  T.2.1, T.2.2,  P.1  , P.2)
return (Ax = A1 + A2 + A3)
}
TriArea(x1, y1, x2, y2, x3, y3){
return Abs((x1 * (y2-y3) + x2 * (y3-y1) + x3 * (y1-y2)) / 2)
}
```
Output:
```[0, 0]  is within triangle?	ture
[0, 1]  is within triangle?	ture
[3, 1]  is within triangle?	false
[5.4142857, 14.349206]  is within triangle?	false```

## C

Translation of: Go
```#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>

const double EPS = 0.001;
const double EPS_SQUARE = 0.000001;

double side(double x1, double y1, double x2, double y2, double x, double y) {
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1);
}

bool naivePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double checkSide1 = side(x1, y1, x2, y2, x, y) >= 0;
double checkSide2 = side(x2, y2, x3, y3, x, y) >= 0;
double checkSide3 = side(x3, y3, x1, y1, x, y) >= 0;
return checkSide1 && checkSide2 && checkSide3;
}

bool pointInTriangleBoundingBox(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double xMin = min(x1, min(x2, x3)) - EPS;
double xMax = max(x1, max(x2, x3)) + EPS;
double yMin = min(y1, min(y2, y3)) - EPS;
double yMax = max(y1, max(y2, y3)) + EPS;
return !(x < xMin || xMax < x || y < yMin || yMax < y);
}

double distanceSquarePointToSegment(double x1, double y1, double x2, double y2, double x, double y) {
double p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1);
double dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength;
if (dotProduct < 0) {
return (x - x1) * (x - x1) + (y - y1) * (y - y1);
} else if (dotProduct <= 1) {
double p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y);
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength;
} else {
return (x - x2) * (x - x2) + (y - y2) * (y - y2);
}
}

bool accuratePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
if (!pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)) {
return false;
}
if (naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
return true;
}
if (distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE) {
return true;
}
return false;
}

void printPoint(double x, double y) {
printf("(%f, %f)", x, y);
}

void printTriangle(double x1, double y1, double x2, double y2, double x3, double y3) {
printf("Triangle is [");
printPoint(x1, y1);
printf(", ");
printPoint(x2, y2);
printf(", ");
printPoint(x3, y3);
printf("] \n");
}

void test(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
printTriangle(x1, y1, x2, y2, x3, y3);
printf("Point ");
printPoint(x, y);
printf(" is within triangle? ");
if (accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
printf("true\n");
} else {
printf("false\n");
}
}

int main() {
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 0);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 1);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 3, 1);
printf("\n");

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, 25, 11.11111111111111, 5.414285714285714, 14.349206349206348);
printf("\n");

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, -12.5, 16.666666666666668, 5.414285714285714, 14.349206349206348);
printf("\n");

return 0;
}
```
Output:
```Triangle is [(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (0.000000, 0.000000) is within triangle? true
Triangle is [(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (0.000000, 1.000000) is within triangle? true
Triangle is [(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (3.000000, 1.000000) is within triangle? false

Triangle is [(0.100000, 0.111111), (12.500000, 33.333333), (25.000000, 11.111111)]
Point (5.414286, 14.349206) is within triangle? true

Triangle is [(0.100000, 0.111111), (12.500000, 33.333333), (-12.500000, 16.666667)]
Point (5.414286, 14.349206) is within triangle? true```

## C++

Translation of: C
```#include <iostream>

const double EPS = 0.001;
const double EPS_SQUARE = EPS * EPS;

double side(double x1, double y1, double x2, double y2, double x, double y) {
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1);
}

bool naivePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double checkSide1 = side(x1, y1, x2, y2, x, y) >= 0;
double checkSide2 = side(x2, y2, x3, y3, x, y) >= 0;
double checkSide3 = side(x3, y3, x1, y1, x, y) >= 0;
return checkSide1 && checkSide2 && checkSide3;
}

bool pointInTriangleBoundingBox(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double xMin = std::min(x1, std::min(x2, x3)) - EPS;
double xMax = std::max(x1, std::max(x2, x3)) + EPS;
double yMin = std::min(y1, std::min(y2, y3)) - EPS;
double yMax = std::max(y1, std::max(y2, y3)) + EPS;
return !(x < xMin || xMax < x || y < yMin || yMax < y);
}

double distanceSquarePointToSegment(double x1, double y1, double x2, double y2, double x, double y) {
double p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1);
double dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength;
if (dotProduct < 0) {
return (x - x1) * (x - x1) + (y - y1) * (y - y1);
} else if (dotProduct <= 1) {
double p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y);
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength;
} else {
return (x - x2) * (x - x2) + (y - y2) * (y - y2);
}
}

bool accuratePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
if (!pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)) {
return false;
}
if (naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
return true;
}
if (distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE) {
return true;
}
return false;
}

void printPoint(double x, double y) {
std::cout << '(' << x << ", " << y << ')';
}

void printTriangle(double x1, double y1, double x2, double y2, double x3, double y3) {
std::cout << "Triangle is [";
printPoint(x1, y1);
std::cout << ", ";
printPoint(x2, y2);
std::cout << ", ";
printPoint(x3, y3);
std::cout << "]\n";
}

void test(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
printTriangle(x1, y1, x2, y2, x3, y3);
std::cout << "Point ";
printPoint(x, y);
std::cout << " is within triangle? ";
if (accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
std::cout << "true\n";
} else {
std::cout << "false\n";
}
}

int main() {
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 0);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 1);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 3, 1);
std::cout << '\n';

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, 25, 11.11111111111111, 5.414285714285714, 14.349206349206348);
std::cout << '\n';

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, -12.5, 16.666666666666668, 5.414285714285714, 14.349206349206348);
std::cout << '\n';

return 0;
}
```
Output:
```Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 0) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 1) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3, 1) is within triangle? false

Triangle is [(0.1, 0.111111), (12.5, 33.3333), (25, 11.1111)]
Point (5.41429, 14.3492) is within triangle? true

Triangle is [(0.1, 0.111111), (12.5, 33.3333), (-12.5, 16.6667)]
Point (5.41429, 14.3492) is within triangle? true```

## Common Lisp

```; There are different algorithms to solve this problem, such as adding areas, adding angles, etc... but these
; solutions are sensitive to rounding errors intrinsic to float operations. We want to avoid these issues, therefore we
; use the following algorithm which only uses multiplication and subtraction: we consider one side of the triangle
; and see on which side of it is the point P located. We can give +1 if it is on the right hand side, -1 for the
; left side, or 0 if it is on the line. If the point is located on the same side relative to all three sides of the triangle
; then the point is inside of it. This has an added advantage that it can be scaled up to other more complicated figures
; (even concave ones, with some minor modifications).

(defun point-inside-triangle (P A B C)
"Is the point P inside the triangle formed by ABC?"
(= (side-of-line P A B)
(side-of-line P B C)
(side-of-line P C A) ))

; This is the version to include those points which are on one of the sides
(defun point-inside-or-on-triangle (P A B C)
"Is the point P inside the triangle formed by ABC or on one of the sides?"
(apply #'= (remove 0 (list (side-of-line P A B) (side-of-line P B C) (side-of-line P C A)))) )

(defun side-of-line (P A B)
"Return +1 if it is on the right side, -1 for the left side, or 0 if it is on the line"
; We use the sign of the determinant of vectors (AB,AM), where M(X,Y) is the query point:
; position = sign((Bx - Ax) * (Y - Ay) - (By - Ay) * (X - Ax))
(signum (- (* (- (car B) (car A))
(- (cdr P) (cdr A)) )
(* (- (cdr B) (cdr A))
(- (car P) (car A)) ))))
```
Output:
```(point-inside-triangle '(0 . 0) '(1.5 . 2.4) '(5.1 . -3.1) '(-3.8 . 1.2))
T

(point-inside-triangle '(0 . 1) '(1.5 . 2.4) '(5.1 . -3.1) '(-3.8 . 1.2))
T

(point-inside-triangle '(3 . 1) '(1.5 . 2.4) '(5.1 . -3.1) '(-3.8 . 1.2))
NIL

(point-inside-triangle '(5.414286 . 14.349206) '(0.1 . 0.111111) '(12.5 . 33.333333) '(25.0 . 11.111111))
T

(point-inside-triangle '(5.414286 . 14.349206) '(0.1 . 0.111111) '(12.5 . 33.333333) '(-12.5 . 16.666667))
NIL
```

## D

Translation of: C++
```import std.algorithm; //.comparison for min and max
import std.stdio;

immutable EPS = 0.001;
immutable EPS_SQUARE = EPS * EPS;

double side(double x1, double y1, double x2, double y2, double x, double y) {
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1);
}

bool naivePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double checkSide1 = side(x1, y1, x2, y2, x, y) >= 0;
double checkSide2 = side(x2, y2, x3, y3, x, y) >= 0;
double checkSide3 = side(x3, y3, x1, y1, x, y) >= 0;
return checkSide1 && checkSide2 && checkSide3;
}

bool pointInTriangleBoundingBox(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double xMin = min(x1, x2, x3) - EPS;
double xMax = max(x1, x2, x3) + EPS;
double yMin = min(y1, y2, y3) - EPS;
double yMax = max(y1, y2, y3) + EPS;
return !(x < xMin || xMax < x || y < yMin || yMax < y);
}

double distanceSquarePointToSegment(double x1, double y1, double x2, double y2, double x, double y) {
double p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1);
double dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength;
if (dotProduct < 0) {
return (x - x1) * (x - x1) + (y - y1) * (y - y1);
} else if (dotProduct <= 1) {
double p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y);
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength;
} else {
return (x - x2) * (x - x2) + (y - y2) * (y - y2);
}
}

bool accuratePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
if (!pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)) {
return false;
}
if (naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
return true;
}
if (distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE) {
return true;
}
return false;
}

void printPoint(double x, double y) {
write('(', x, ", ", y, ')');
}

void printTriangle(double x1, double y1, double x2, double y2, double x3, double y3) {
write("Triangle is [");
printPoint(x1, y1);
write(", ");
printPoint(x2, y2);
write(", ");
printPoint(x3, y3);
writeln(']');
}

void test(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
printTriangle(x1, y1, x2, y2, x3, y3);
write("Point ");
printPoint(x, y);
write(" is within triangle? ");
writeln(accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y));
}

void main() {
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 0);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 1);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 3, 1);
writeln;

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, 25, 11.11111111111111, 5.414285714285714, 14.349206349206348);
writeln;

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, -12.5, 16.666666666666668, 5.414285714285714, 14.349206349206348);
writeln;
}
```
Output:
```Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 0) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 1) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3, 1) is within triangle? false

Triangle is [(0.1, 0.111111), (12.5, 33.3333), (25, 11.1111)]
Point (5.41429, 14.3492) is within triangle? true

Triangle is [(0.1, 0.111111), (12.5, 33.3333), (-12.5, 16.6667)]
Point (5.41429, 14.3492) is within triangle? true```

## Dart

Translation of: C++
```import 'dart:math';

const double EPS = 0.001;
const double EPS_SQUARE = EPS * EPS;

double side(double x1, double y1, double x2, double y2, double x, double y) {
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1);
}

bool naivePointInTriangle(double x1, double y1, double x2, double y2, double x3,
double y3, double x, double y) {
double checkSide1 = side(x1, y1, x2, y2, x, y); // >= 0;
double checkSide2 = side(x2, y2, x3, y3, x, y); // >= 0;
double checkSide3 = side(x3, y3, x1, y1, x, y); // >= 0;
if (checkSide1 >= 0 && checkSide2 >= 0 && checkSide3 >= 0) {
return true;
} else {
return false;
}
}

bool pointInTriangleBoundingBox(double x1, double y1, double x2, double y2,
double x3, double y3, double x, double y) {
double xMin = min(x1, min(x2, x3)) - EPS;
double xMax = max(x1, max(x2, x3)) + EPS;
double yMin = min(y1, min(y2, y3)) - EPS;
double yMax = max(y1, max(y2, y3)) + EPS;
return !(x < xMin || xMax < x || y < yMin || yMax < y);
}

double distanceSquarePointToSegment(
double x1, double y1, double x2, double y2, double x, double y) {
double p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1);
double dotProduct =
((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength;
if (dotProduct < 0) {
return (x - x1) * (x - x1) + (y - y1) * (y - y1);
} else if (dotProduct <= 1) {
double p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y);
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength;
} else {
return (x - x2) * (x - x2) + (y - y2) * (y - y2);
}
}

bool accuratePointInTriangle(double x1, double y1, double x2, double y2,
double x3, double y3, double x, double y) {
if (!pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)) {
return false;
}
if (naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
return true;
}
if (distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE) {
return true;
}
return false;
}

void printTriangle(
double x1, double y1, double x2, double y2, double x3, double y3) {
print("Triangle is [(\$x1, \$y1), (\$x2, \$y2), (\$x3, \$y3)]");
}

void test(double x1, double y1, double x2, double y2, double x3, double y3,
double x, double y) {
printTriangle(x1, y1, x2, y2, x3, y3);
print("Point (\$x, \$y) is within triangle? ");
if (accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
print("true");
} else {
print("false");
}
}

void main() {
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 0);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 1);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 3, 1);
print('');

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, 25, 11.11111111111111,
5.414285714285714, 14.349206349206348);
print('');

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, -12.5,
16.666666666666668, 5.414285714285714, 14.349206349206348);
print('');
}
```

## Factor

Uses the parametric equations method from [5].

```USING: accessors fry io kernel locals math math.order sequences ;

TUPLE: point x y ;
C: <point> point
: >point< ( point -- x y ) [ x>> ] [ y>> ] bi ;

TUPLE: triangle p1 p2 p3 ;
C: <triangle> triangle
: >triangle< ( triangle -- x1 y1 x2 y2 x3 y3 )
[ p1>> ] [ p2>> ] [ p3>> ] tri [ >point< ] tri@ ;

:: point-in-triangle? ( point triangle -- ? )
point >point< triangle >triangle< :> ( x y x1 y1 x2 y2 x3 y3 )
y2 y3 - x1 * x3 x2 - y1 * + x2 y3 * + y2 x3 * - :> d
y3 y1 - x * x1 x3 - y * + x1 y3 * - y1 x3 * + d / :> t1
y2 y1 - x * x1 x2 - y * + x1 y2 * - y1 x2 * + d neg / :> t2
t1 t2 + :> s

t1 t2 [ 0 1 between? ] bi@ and s 1 <= and ;

! Test if it works.

20 <iota> dup [ swap <point> ] cartesian-map                     ! Make a matrix of points
3 3 <point> 16 10 <point> 10 16 <point> <triangle>               ! Make a triangle
'[ [ _ point-in-triangle? "#" "." ? write ] each nl ] each nl    ! Show points inside the triangle with '#'
```
Output:
```....................
....................
....................
...#................
....#...............
.....##.............
.....####...........
......#####.........
......#######.......
.......########.....
.......##########...
........########....
........#######.....
.........#####......
.........####.......
..........##........
..........#.........
....................
....................
....................
```

## Fortran

```PROGRAM POINT_WITHIN_TRIANGLE

IMPLICIT NONE

REAL (KIND = SELECTED_REAL_KIND (8)) px, py, ax, ay, bx, by, cx, cy

px = 0.0
py = 0.0
ax = 1.5
ay = 2.4
bx = 5.1
by = -3.1
cx = -3.8
cy = 1.2

IF (IS_P_IN_ABC (px, py, ax, ay, bx, by, cx, cy)) THEN

WRITE (*, *) 'Point (', px, ', ', py, ') is within triangle &
[(', ax, ', ', ay,'), (', bx, ', ', by, '), (', cx, ', ', cy, ')].'

ELSE

WRITE (*, *) 'Point (', px, ', ', py, ') is not within triangle &
[(', ax, ', ', ay,'), (', bx, ', ', by, '), (', cx, ', ', cy, ')].'

END IF

CONTAINS

!Provide xy values of points P, A, B, C, respectively.
LOGICAL FUNCTION IS_P_IN_ABC (px, py, ax, ay, bx, by, cx, cy)

REAL (KIND = SELECTED_REAL_KIND (8)), INTENT (IN) :: px, py, ax, ay, bx, by, cx, cy
REAL (KIND = SELECTED_REAL_KIND (8)) :: vabx, vaby, vacx, vacy, a, b

vabx = bx - ax
vaby = by - ay
vacx = cx - ax
vacy = cy - ay

a = ((px * vacy - py * vacx) - (ax * vacy - ay * vacx)) / &
(vabx * vacy - vaby * vacx)
b = -((px * vaby - py * vabx) - (ax * vaby - ay * vabx)) / &
(vabx * vacy - vaby * vacx)

IF ((a .GT. 0) .AND. (b .GT. 0) .AND. (a + b < 1)) THEN

IS_P_IN_ABC = .TRUE.

ELSE

IS_P_IN_ABC = .FALSE.

END IF

END FUNCTION IS_P_IN_ABC

END PROGRAM POINT_WITHIN_TRIANGLE
```
Output:
```
Point (   0.0000000000000000      ,    0.0000000000000000      ) is within triangle [(   1.5000000000000000      ,    2.4000000953674316      ), (   5.0999999046325684      ,   -3.0999999046325684      ), (  -3.7999999523162842      ,    1.2000000476837158      )].

```

## FreeBASIC

```type p2d
x as double   'define a two-dimensional point
y as double
end type

function in_tri( A as p2d, B as p2d, C as p2d, P as p2d ) as boolean
'uses barycentric coordinates to determine if point P is inside
'the triangle defined by points A, B, C
dim as double AreaD = (-B.y*C.x + A.y*(-B.x + C.x) + A.x*(B.y - C.y) + B.x*C.y)
dim as double s = (A.y*C.x - A.x*C.y + (C.y - A.y)*P.x + (A.x - C.x)*P.y)/AreaD
dim as double t = (A.x*B.y - A.y*B.x + (A.y - B.y)*P.x + (B.x - A.x)*P.y)/AreaD
if s<=0 then return false
if t<=0 then return false
if s+t>=1 then return false
return true
end function

dim as p2d A,B,C,P    'generate some arbitrary triangle
A.x = 4.14 : A.y = -1.12
B.x = 8.1 : B.y =-4.9
C.x = 1.5: C.y = -9.3

for y as double = -0.25 to -9.75 step -0.5    'display a 10x10 square
for x as double = 0.125 to 9.875 step 0.25
P.x = x : P.y = y
if in_tri(A,B,C,P) then print "@"; else print ".";   'with all the points inside the triangle indicated
next x
print
next y
```
Output:
```........................................
........................................
................@.......................
................@@@.....................
...............@@@@@@...................
..............@@@@@@@@@.................
..............@@@@@@@@@@@...............
.............@@@@@@@@@@@@@@@............
.............@@@@@@@@@@@@@@@@@..........
............@@@@@@@@@@@@@@@@@@@@........
...........@@@@@@@@@@@@@@@@@@@..........
...........@@@@@@@@@@@@@@@@.............
..........@@@@@@@@@@@@@@................
.........@@@@@@@@@@@@...................
.........@@@@@@@@@......................
........@@@@@@@.........................
.......@@@@@............................
.......@@...............................
........................................
........................................

```

## FutureBasic

```_window = 1
begin enum 1
_textLabel
end enum

void local fn BuildWindow
'~'1
window _window, @"Find if a point is within a triangle", (0, 0, 340, 360 )
WindowCenter(_window)
WindowSubclassContentView(_window)
ViewSetFlipped( _windowContentViewTag, YES )
ViewSetNeedsDisplay( _windowContentViewTag )

subclass textLabel _textLabel, @"", ( 20, 320, 300, 20 ), _window
end fn

void local fn DrawInView( tag as NSInteger )
'~'1
BezierPathRef path = fn BezierPathInit
BezierPathMoveToPoint( path, fn CGPointMake(  30, 300 ) )
BezierPathLineToPoint( path, fn CGPointMake( 300, 300 ) )
BezierPathLineToPoint( path, fn CGPointMake( 150,  30 ) )
BezierPathClose( path )
BezierPathStrokeFill( path, 3.0, fn ColorBlack, fn ColorGreen )
AppSetProperty( @"path", path )
end fn

void local fn DoMouse( tag as NSInteger )
'~'1
CGPoint pt = fn EventLocationInView( tag )
if ( fn BezierPathContainsPoint( fn AppProperty( @"path" ), pt ) )
ControlSetStringValue( _textLabel, fn StringWithFormat( @"Inside triangle:  x = %.f  y = %.f", pt.x, pt.y ) )
else
ControlSetStringValue( _textLabel, fn StringWithFormat( @"Outside triangle:  x = %.f  y = %.f", pt.x, pt.y ) )
end if
end fn

void local fn DoDialog( ev as long, tag as long )
'~'1
select ( ev )
case _viewDrawRect   : fn DrawInView(tag)
case _viewMouseDown  : fn DoMouse( tag )
case _viewMouseMoved : fn DoMouse( tag )
end select
end fn

fn BuildWindow

on dialog fn DoDialog

HandleEvents```
Output:

## Go

Translation of: Wren
```package main

import (
"fmt"
"math"
)

const EPS = 0.001
const EPS_SQUARE = EPS * EPS

func side(x1, y1, x2, y2, x, y float64) float64 {
return (y2-y1)*(x-x1) + (-x2+x1)*(y-y1)
}

func naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y float64) bool {
checkSide1 := side(x1, y1, x2, y2, x, y) >= 0
checkSide2 := side(x2, y2, x3, y3, x, y) >= 0
checkSide3 := side(x3, y3, x1, y1, x, y) >= 0
return checkSide1 && checkSide2 && checkSide3
}

func pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y float64) bool {
xMin := math.Min(x1, math.Min(x2, x3)) - EPS
xMax := math.Max(x1, math.Max(x2, x3)) + EPS
yMin := math.Min(y1, math.Min(y2, y3)) - EPS
yMax := math.Max(y1, math.Max(y2, y3)) + EPS
return !(x < xMin || xMax < x || y < yMin || yMax < y)
}

func distanceSquarePointToSegment(x1, y1, x2, y2, x, y float64) float64 {
p1_p2_squareLength := (x2-x1)*(x2-x1) + (y2-y1)*(y2-y1)
dotProduct := ((x-x1)*(x2-x1) + (y-y1)*(y2-y1)) / p1_p2_squareLength
if dotProduct < 0 {
return (x-x1)*(x-x1) + (y-y1)*(y-y1)
} else if dotProduct <= 1 {
p_p1_squareLength := (x1-x)*(x1-x) + (y1-y)*(y1-y)
return p_p1_squareLength - dotProduct*dotProduct*p1_p2_squareLength
} else {
return (x-x2)*(x-x2) + (y-y2)*(y-y2)
}
}

func accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y float64) bool {
if !pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y) {
return false
}
if naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y) {
return true
}
if distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE {
return true
}
if distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE {
return true
}
if distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE {
return true
}
return false
}

func main() {
pts := [][2]float64{{0, 0}, {0, 1}, {3, 1}}
tri := [][2]float64{{3.0 / 2, 12.0 / 5}, {51.0 / 10, -31.0 / 10}, {-19.0 / 5, 1.2}}
fmt.Println("Triangle is", tri)
x1, y1 := tri[0][0], tri[0][1]
x2, y2 := tri[1][0], tri[1][1]
x3, y3 := tri[2][0], tri[2][1]
for _, pt := range pts {
x, y := pt[0], pt[1]
within := accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
fmt.Println("Point", pt, "is within triangle?", within)
}
fmt.Println()
tri = [][2]float64{{1.0 / 10, 1.0 / 9}, {100.0 / 8, 100.0 / 3}, {100.0 / 4, 100.0 / 9}}
fmt.Println("Triangle is", tri)
x1, y1 = tri[0][0], tri[0][1]
x2, y2 = tri[1][0], tri[1][1]
x3, y3 = tri[2][0], tri[2][1]
x := x1 + (3.0/7)*(x2-x1)
y := y1 + (3.0/7)*(y2-y1)
pt := [2]float64{x, y}
within := accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
fmt.Println("Point", pt, "is within triangle ?", within)
fmt.Println()
tri = [][2]float64{{1.0 / 10, 1.0 / 9}, {100.0 / 8, 100.0 / 3}, {-100.0 / 8, 100.0 / 6}}
fmt.Println("Triangle is", tri)
x3 = tri[2][0]
y3 = tri[2][1]
within = accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
fmt.Println("Point", pt, "is within triangle ?", within)
}
```
Output:
```Triangle is [[1.5 2.4] [5.1 -3.1] [-3.8 1.2]]
Point [0 0] is within triangle? true
Point [0 1] is within triangle? true
Point [3 1] is within triangle? false

Triangle is [[0.1 0.1111111111111111] [12.5 33.333333333333336] [25 11.11111111111111]]
Point [5.414285714285714 14.349206349206348] is within triangle ? true

Triangle is [[0.1 0.1111111111111111] [12.5 33.333333333333336] [-12.5 16.666666666666668]]
Point [5.414285714285714 14.349206349206348] is within triangle ? true
```

## GW-BASIC

```10 PIT1X! = 3 : PIT1Y! = 1.3 :    REM arbitrary triangle for demonstration
20 PIT2X! = 17.222 : PIT2Y! = 10
30 PIT3X! =  5.5 : PIT3Y! = 18.212
40 FOR PITPY! = 0 TO 19 STEP 1
50 FOR PITPX! = 0 TO 20 STEP .5
60 GOSUB 1000
70 IF PITRES% = 0 THEN PRINT "."; ELSE PRINT "#";
80 NEXT PITPX!
90 PRINT
100 NEXT PITPY!
110 END
1000 REM Detect if point is in triangle. Takes 8 double-precision
1010 REM values: (PIT1X!, PIT1Y!), (PIT2X!, PIT2Y!), (PIT3X!, PIT3Y!)
1020 REM for the coordinates of the corners of the triangle
1030 REM and (PITPX!, PITPY!) for the coordinates of the test point
1040 REM Returns PITRES%: 1=in triangle, 0=not in it
1050 PITDAR! = -PIT2Y!*PIT3X! + PIT1Y!*(-PIT2X! + PIT3X!) + PIT1X!*(PIT2Y - PIT3Y!) + PIT2X!*PIT3Y!
1060 PITXXS = (PIT1Y!*PIT3X! - PIT1X!*PIT3Y! + (PIT3Y! - PIT1Y!)*PITPX! + (PIT1X! - PIT3X!)*PITPY!)/PITDAR!
1070 PITXXT = (PIT1X!*PIT2Y! - PIT1Y!*PIT2X! + (PIT1Y! - PIT2Y!)*PITPX! + (PIT2X! - PIT1X!)*PITPY!)/PITDAR!
1080 PITRES% = 0
1090 IF PITXXS!<=0 THEN RETURN
1100 IF PITXXT!<=0 THEN RETURN
1110 IF PITXXS!+PITXXT!>=1 THEN RETURN
1120 PITRES% = 1
1130 RETURN
```
Output:
```.........................................
.........................................
.......##................................
.......#####.............................
.......########..........................
........###########......................
........##############...................
........#################................
........####################.............
.........#######################.........
.........##########################......
.........#######################.........
..........###################............
..........################...............
..........##############.................
...........##########....................
...........#######.......................
...........####..........................
...........#.............................
.........................................

```

## Haskell

The point to be tested is transformed by affine transformation which turns given triangle to the simplex: Triangle (0,0) (0,s) (s,0), where s is half of the triangles' area. After that criteria of overlapping become trivial. Affinity allows to avoid division, so all functions work for points on the integer, or rational, or even modular meshes as well.

```type Pt a = (a, a)

data Overlapping = Inside | Outside | Boundary
deriving (Show, Eq)

data Triangle a = Triangle (Pt a) (Pt a) (Pt a)
deriving Show

vertices (Triangle a b c) = [a, b, c]

-- Performs the affine transformation
-- which turns a triangle to Triangle (0,0) (0,s) (s,0)
-- where s is half of the triangles' area
toTriangle :: Num a => Triangle a -> Pt a -> (a, Pt a)
toTriangle t (x,y) = let
[(x0,y0), (x1,y1), (x2,y2)] = vertices t
s = x2*(y0-y1)+x0*(y1-y2)+x1*(-y0+y2)
in  ( abs s
, ( signum s * (x2*(-y+y0)+x0*(y-y2)+x*(-y0+y2))
, signum s * (x1*(y-y0)+x*(y0-y1)+x0*(-y+y1))))

overlapping :: (Eq a, Ord a, Num a) =>
Triangle a -> Pt a -> Overlapping
overlapping t p =  case toTriangle t p of
(s, (x, y))
| s == 0 && (x == 0 || y == 0)     -> Boundary
| s == 0                           -> Outside
| x > 0 && y > 0 && y < s - x      -> Inside
| (x <= s && x >= 0) &&
(y <= s && y >= 0) &&
(x == 0 || y == 0 || y == s - x) -> Boundary
| otherwise                        -> Outside
```

Testing

```tests = let
t1 = Triangle (2,0) (-1,2) (-2,-2)
bs = [(2,0), (-1,2), (-2,-2), (0,-1), (1/2,1), (-3/2,0)]
is = [(0,0), (0,1), (-1,0), (-1,1), (-1,-1)]
os = [(1,1), (-2,2), (100,100), (2.00000001, 0)]

t2 = Triangle (1,2) (1,2) (-1,3)
ps = [(1,2), (0,5/2), (0,2), (1,3)]

in mapM_ print [ overlapping t1 <\$> bs
, overlapping t1 <\$> is
, overlapping t1 <\$> os
, overlapping t2 <\$> ps]
test2 = unlines
[ [case overlapping t (i,j) of
Inside -> '∗'
Boundary -> '+'
Outside -> '·'
| i <- [-10..10] :: [Int] ]
| j <- [-5..5] :: [Int] ]
where t = Triangle (-8,-3) (8,1) (-1,4)
```
```λ> tests
[Boundary,Boundary,Boundary,Boundary,Boundary,Boundary]
[Inside,Inside,Inside,Inside,Inside]
[Outside,Outside,Outside,Outside]
[Boundary,Boundary,Outside,Outside]

λ> putStrLn test2
·····················
·····················
··+··················
···+∗∗+··············
····+∗∗∗∗∗+··········
·····+∗∗∗∗∗∗∗∗+······
······+∗∗∗∗∗∗∗∗∗∗∗+··
·······+∗∗∗∗∗∗∗+·····
········+∗∗∗+········
·········+···········
·····················```

## J

Implementation, using complex numbers to represent x,y coordinates:

```area=: [:| 0.5-/ .*@,.+. NB. signed area of triangle
I3=: =i.3 NB. identity matrix
inside=: {{ (area y)=+/area"1|:(I3*x)+(1-I3)*y }}
```

This is based on the algorithm documented for the ada implementation: compute the area of triangles using the determinant method (we want the absolute area here), and check whether the triangles formed with the test point and the sides of the test triangle matches the area of the test triangle.

Examples:
```   0j0 inside 1.5j2.4 5.1j_3.1 _3.8j1.2
1
0j1 inside 1.5j2.4 5.1j_3.1 _3.8j1.2
1
3j1 inside 1.5j2.4 5.1j_3.1 _3.8j1.2
0
5.414285714285714j14.349206349206348 inside 0.1j1r9 12.5j100r3 25j100r9
1
5.414285714285714j14.349206349206348 inside 0.1j1r9 12.5j100r3 _12.5j100r6
1
```

## Java

Translation of: Go
```import java.util.Objects;

public class FindTriangle {
private static final double EPS = 0.001;
private static final double EPS_SQUARE = EPS * EPS;

public static class Point {
private final double x, y;

public Point(double x, double y) {
this.x = x;
this.y = y;
}

public double getX() {
return x;
}

public double getY() {
return y;
}

@Override
public String toString() {
return String.format("(%f, %f)", x, y);
}
}

public static class Triangle {
private final Point p1, p2, p3;

public Triangle(Point p1, Point p2, Point p3) {
this.p1 = Objects.requireNonNull(p1);
this.p2 = Objects.requireNonNull(p2);
this.p3 = Objects.requireNonNull(p3);
}

public Point getP1() {
return p1;
}

public Point getP2() {
return p2;
}

public Point getP3() {
return p3;
}

private boolean pointInTriangleBoundingBox(Point p) {
var xMin = Math.min(p1.getX(), Math.min(p2.getX(), p3.getX())) - EPS;
var xMax = Math.max(p1.getX(), Math.max(p2.getX(), p3.getX())) + EPS;
var yMin = Math.min(p1.getY(), Math.min(p2.getY(), p3.getY())) - EPS;
var yMax = Math.max(p1.getY(), Math.max(p2.getY(), p3.getY())) + EPS;
return !(p.getX() < xMin || xMax < p.getX() || p.getY() < yMin || yMax < p.getY());
}

private static double side(Point p1, Point p2, Point p) {
return (p2.getY() - p1.getY()) * (p.getX() - p1.getX()) + (-p2.getX() + p1.getX()) * (p.getY() - p1.getY());
}

private boolean nativePointInTriangle(Point p) {
boolean checkSide1 = side(p1, p2, p) >= 0;
boolean checkSide2 = side(p2, p3, p) >= 0;
boolean checkSide3 = side(p3, p1, p) >= 0;
return checkSide1 && checkSide2 && checkSide3;
}

private double distanceSquarePointToSegment(Point p1, Point p2, Point p) {
double p1_p2_squareLength = (p2.getX() - p1.getX()) * (p2.getX() - p1.getX()) + (p2.getY() - p1.getY()) * (p2.getY() - p1.getY());
double dotProduct = ((p.getX() - p1.getX()) * (p2.getX() - p1.getX()) + (p.getY() - p1.getY()) * (p2.getY() - p1.getY())) / p1_p2_squareLength;
if (dotProduct < 0) {
return (p.getX() - p1.getX()) * (p.getX() - p1.getX()) + (p.getY() - p1.getY()) * (p.getY() - p1.getY());
}
if (dotProduct <= 1) {
double p_p1_squareLength = (p1.getX() - p.getX()) * (p1.getX() - p.getX()) + (p1.getY() - p.getY()) * (p1.getY() - p.getY());
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength;
}
return (p.getX() - p2.getX()) * (p.getX() - p2.getX()) + (p.getY() - p2.getY()) * (p.getY() - p2.getY());
}

private boolean accuratePointInTriangle(Point p) {
if (!pointInTriangleBoundingBox(p)) {
return false;
}
if (nativePointInTriangle(p)) {
return true;
}
if (distanceSquarePointToSegment(p1, p2, p) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(p2, p3, p) <= EPS_SQUARE) {
return true;
}
return distanceSquarePointToSegment(p3, p1, p) <= EPS_SQUARE;
}

public boolean within(Point p) {
Objects.requireNonNull(p);
return accuratePointInTriangle(p);
}

@Override
public String toString() {
return String.format("Triangle[%s, %s, %s]", p1, p2, p3);
}
}

private static void test(Triangle t, Point p) {
System.out.println(t);
System.out.printf("Point %s is within triangle? %s\n", p, t.within(p));
}

public static void main(String[] args) {
var p1 = new Point(1.5, 2.4);
var p2 = new Point(5.1, -3.1);
var p3 = new Point(-3.8, 1.2);
var tri = new Triangle(p1, p2, p3);
test(tri, new Point(0, 0));
test(tri, new Point(0, 1));
test(tri, new Point(3, 1));
System.out.println();

p1 = new Point(1.0 / 10, 1.0 / 9);
p2 = new Point(100.0 / 8, 100.0 / 3);
p3 = new Point(100.0 / 4, 100.0 / 9);
tri = new Triangle(p1, p2, p3);
var pt = new Point(p1.getX() + (3.0 / 7) * (p2.getX() - p1.getX()), p1.getY() + (3.0 / 7) * (p2.getY() - p1.getY()));
test(tri, pt);
System.out.println();

p3 = new Point(-100.0 / 8, 100.0 / 6);
tri = new Triangle(p1, p2, p3);
test(tri, pt);
}
}
```
Output:
```Triangle[(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (0.000000, 0.000000) is within triangle? true
Triangle[(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (0.000000, 1.000000) is within triangle? true
Triangle[(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (3.000000, 1.000000) is within triangle? false

Triangle[(0.100000, 0.111111), (12.500000, 33.333333), (25.000000, 11.111111)]
Point (5.414286, 14.349206) is within triangle? true

Triangle[(0.100000, 0.111111), (12.500000, 33.333333), (-12.500000, 16.666667)]
Point (5.414286, 14.349206) is within triangle? true```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

Adapted from Wren

A point is represented by [x,y] and denoted by P1, P2, P3, or Q.

A triangle is represented by an array of points: [P1, P2, P3].

Preliminaries

```def sum_of_squares(stream): reduce stream as \$x (0; . + \$x * \$x);

def distanceSquared(P1; P2): sum_of_squares(P1[0]-P2[0], P1[1]-P2[1]);

# Emit {x1,y1, ...} for the input triangle
def xy:
{ x1: .[0][0],
y1: .[0][1],
x2: .[1][0],
y2: .[1][1],
x3: .[2][0],
y3: .[2][1] };

def EPS: 0.001;
def EPS_SQUARE: EPS * EPS;```
```def side(P1; P2; Q):
[P1, P2, Q]
| xy
| (.y2 - .y1)*(.x3 - .x1) + (-.x2 + .x1)*(.y3 - .y1);

def naivePointInTriangle(P1; P2; P3; Q):
side(P1; P2; Q) >= 0
and side(P2; P3; Q) >= 0
and side(P3; P1; Q) >= 0;

def pointInTriangleBoundingBox(P1; P2; P3; Q):
[P1,P2,P3]
| (map(.[0]) | min - EPS) as \$xMin
| (map(.[0]) | max + EPS) as \$xMax
| (map(.[1]) | min - EPS) as \$yMin
| (map(.[1]) | max + EPS) as \$yMax
| (Q[0] < \$xMin or \$xMax < Q[0] or Q[1] < \$yMin or \$yMax < Q[1]) | not;

def distanceSquarePointToSegment(P1; P2; Q):
distanceSquared(P1; P2) as \$p1_p2_squareLength
| [P1, P2, Q]
| xy
| (((.x3 - .x1)*(.x2 - .x1) + (.y3 - .y1)*(.y2 - .y1)) / \$p1_p2_squareLength) as \$dotProduct
| if \$dotProduct < 0
then sum_of_squares(.x3 - .x1, .y3 - .y1)
elif \$dotProduct <= 1
then sum_of_squares(.x1 - .x3, .y1 - .y3) as \$p_p1_squareLength
|  \$p_p1_squareLength - \$dotProduct * \$dotProduct * \$p1_p2_squareLength
else sum_of_squares(.x3 - .x2, .y3 - .y2)
end;

def accuratePointInTriangle(P1; P2; P3; Q):
if (pointInTriangleBoundingBox(P1; P2; P3; Q) | not)     then false
elif naivePointInTriangle(P1; P2; P3; Q)                   then true
elif distanceSquarePointToSegment(P1; P2; Q) <= EPS_SQUARE then true
elif distanceSquarePointToSegment(P2; P3; Q) <= EPS_SQUARE then true
elif distanceSquarePointToSegment(P3; P1; Q) <= EPS_SQUARE then true
else false
end;```

Examples

```def task1:
def pts: [ [0, 0], [0, 1], [3, 1]];
"Triangle is \(.)",
(. as [\$P1, \$P2, \$P3]
| pts[] as \$Q
| accuratePointInTriangle(\$P1; \$P2; \$P3; \$Q) as \$within
| "Point \(\$Q) is within triangle ? \(\$within)"
);

def task2:
"Triangle is \(.)",
(. as [\$P1, \$P2, \$P3]
| [ \$P1[0] + (3/7)*(\$P2[0] - \$P1[0]), \$P1[1] + (3/7)*(\$P2[1] - \$P1[1]) ] as \$Q
| accuratePointInTriangle(\$P1; \$P2; \$P3; \$Q) as \$within
| "Point \(\$Q) is within triangle ? \(\$within)"
);

([ [3/2, 12/5], [51/10, -31/10], [-19/5,   1.2] ] | task1), "",
([ [1/10, 1/9], [100/8,  100/3], [100/4, 100/9] ] | task2), "",
([ [1/10, 1/9], [100/8, 100/3], [-100/8, 100/6] ] | task2)```
Output:
```Triangle is [[1.5,2.4],[5.1,-3.1],[-3.8,1.2]]
Point [0,0] is within triangle ? true
Point [0,1] is within triangle ? true
Point [3,1] is within triangle ? false

Triangle is [[0.1,0.1111111111111111],[12.5,33.333333333333336],[25,11.11111111111111]]
Point [5.414285714285714,14.349206349206348] is within triangle ? true

Triangle is [[0.1,0.1111111111111111],[12.5,33.333333333333336],[-12.5,16.666666666666668]]
Point [5.414285714285714,14.349206349206348] is within triangle ? true
```

## Julia

Translation of: Python

Using the Wren examples.

```Point(x, y) = [x, y]
Triangle(a, b, c) = [a, b, c]
LEzero(x) =  x < 0 || isapprox(x, 0, atol=0.00000001)
GEzero(x) =  x > 0 || isapprox(x, 0, atol=0.00000001)

""" Determine which side of plane cut by line (p2, p3) p1 is on """
side(p1, p2, p3) = (p1[1] - p3[1]) * (p2[2] - p3[2]) - (p2[1] - p3[1]) * (p1[2] - p3[2])

"""
Determine if point is within triangle formed by points p1, p2, p3.
If so, the point will be on the same side of each of the half planes
defined by vectors p1p2, p2p3, and p3p1. Each z is positive if outside,
negative if inside such a plane. All should be positive or all negative
if point is within the triangle.
"""
function iswithin(point, p1, p2, p3)
z1 = side(point, p1, p2)
z2 = side(point, p2, p3)
z3 = side(point, p3, p1)
notanyneg = GEzero(z1) && GEzero(z2) && GEzero(z3)
notanypos = LEzero(z1) && LEzero(z2) && LEzero(z3)
return notanyneg || notanypos
end

const POINTS = [Point(0 // 1, 0 // 1), Point(0 // 1, 1 // 1), Point(3 // 1, 1 // 1),
Point(1 // 10 + (3 // 7) * (100 // 8 - 1 // 10), 1 // 9 + (3 // 7) * (100 // 3 - 1 // 9)),
Point(3 // 2, 12 // 5), Point(51 // 100, -31 // 100), Point(-19 // 50, 6 // 5),
Point(1 // 10, 1 // 9), Point(25 / 2, 100 // 3), Point(25, 100 // 9),
Point(-25 // 2, 50 // 3)
]

const TRI = [
Triangle(POINTS[5], POINTS[6], POINTS[7]),
Triangle(POINTS[8], POINTS[9], POINTS[10]),
Triangle(POINTS[8], POINTS[9], POINTS[11])
]

for tri in TRI
pstring(pt) = "[\$(Float32(pt[1])), \$(Float32(pt[2]))]"
println("\nUsing triangle [", join([pstring(x) for x in tri], ", "), "]:")
a, b, c = tri[1], tri[2], tri[3]
for p in POINTS[1:4]
isornot = iswithin(p, a, b, c) ? "is" : "is not"
println("Point \$(pstring(p)) \$isornot within the triangle.")
end
end
```
Output:
```Using triangle [[1.5, 2.4], [0.51, -0.31], [-0.38, 1.2]]:
Point [0.0, 0.0] is not within the triangle.
Point [0.0, 1.0] is within the triangle.
Point [3.0, 1.0] is not within the triangle.
Point [5.4142857, 14.349206] is not within the triangle.

Using triangle [[0.1, 0.11111111], [12.5, 33.333332], [25.0, 11.111111]]:
Point [0.0, 0.0] is not within the triangle.
Point [0.0, 1.0] is not within the triangle.
Point [3.0, 1.0] is not within the triangle.
Point [5.4142857, 14.349206] is within the triangle.

Using triangle [[0.1, 0.11111111], [12.5, 33.333332], [-12.5, 16.666666]]:
Point [0.0, 0.0] is not within the triangle.
Point [0.0, 1.0] is within the triangle.
Point [3.0, 1.0] is not within the triangle.
Point [5.4142857, 14.349206] is within the triangle.
```

## Kotlin

Translation of: Java
```import kotlin.math.max
import kotlin.math.min

private const val EPS = 0.001
private const val EPS_SQUARE = EPS * EPS

private fun test(t: Triangle, p: Point) {
println(t)
println("Point \$p is within triangle ? \${t.within(p)}")
}

fun main() {
var p1 = Point(1.5, 2.4)
var p2 = Point(5.1, -3.1)
var p3 = Point(-3.8, 1.2)
var tri = Triangle(p1, p2, p3)
test(tri, Point(0.0, 0.0))
test(tri, Point(0.0, 1.0))
test(tri, Point(3.0, 1.0))
println()
p1 = Point(1.0 / 10, 1.0 / 9)
p2 = Point(100.0 / 8, 100.0 / 3)
p3 = Point(100.0 / 4, 100.0 / 9)
tri = Triangle(p1, p2, p3)
val pt = Point(p1.x + 3.0 / 7 * (p2.x - p1.x), p1.y + 3.0 / 7 * (p2.y - p1.y))
test(tri, pt)
println()
p3 = Point(-100.0 / 8, 100.0 / 6)
tri = Triangle(p1, p2, p3)
test(tri, pt)
}

class Point(val x: Double, val y: Double) {
override fun toString(): String {
return "(\$x, \$y)"
}
}

class Triangle(private val p1: Point, private val p2: Point, private val p3: Point) {
private fun pointInTriangleBoundingBox(p: Point): Boolean {
val xMin = min(p1.x, min(p2.x, p3.x)) - EPS
val xMax = max(p1.x, max(p2.x, p3.x)) + EPS
val yMin = min(p1.y, min(p2.y, p3.y)) - EPS
val yMax = max(p1.y, max(p2.y, p3.y)) + EPS
return !(p.x < xMin || xMax < p.x || p.y < yMin || yMax < p.y)
}

private fun nativePointInTriangle(p: Point): Boolean {
val checkSide1 = side(p1, p2, p) >= 0
val checkSide2 = side(p2, p3, p) >= 0
val checkSide3 = side(p3, p1, p) >= 0
return checkSide1 && checkSide2 && checkSide3
}

private fun distanceSquarePointToSegment(p1: Point, p2: Point, p: Point): Double {
val p1P2SquareLength = (p2.x - p1.x) * (p2.x - p1.x) + (p2.y - p1.y) * (p2.y - p1.y)
val dotProduct = ((p.x - p1.x) * (p2.x - p1.x) + (p.y - p1.y) * (p2.y - p1.y)) / p1P2SquareLength
if (dotProduct < 0) {
return (p.x - p1.x) * (p.x - p1.x) + (p.y - p1.y) * (p.y - p1.y)
}
if (dotProduct <= 1) {
val pP1SquareLength = (p1.x - p.x) * (p1.x - p.x) + (p1.y - p.y) * (p1.y - p.y)
return pP1SquareLength - dotProduct * dotProduct * p1P2SquareLength
}
return (p.x - p2.x) * (p.x - p2.x) + (p.y - p2.y) * (p.y - p2.y)
}

private fun accuratePointInTriangle(p: Point): Boolean {
if (!pointInTriangleBoundingBox(p)) {
return false
}
if (nativePointInTriangle(p)) {
return true
}
if (distanceSquarePointToSegment(p1, p2, p) <= EPS_SQUARE) {
return true
}
return if (distanceSquarePointToSegment(p2, p3, p) <= EPS_SQUARE) {
true
} else distanceSquarePointToSegment(p3, p1, p) <= EPS_SQUARE
}

fun within(p: Point): Boolean {
return accuratePointInTriangle(p)
}

override fun toString(): String {
return "Triangle[\$p1, \$p2, \$p3]"
}

companion object {
private fun side(p1: Point, p2: Point, p: Point): Double {
return (p2.y - p1.y) * (p.x - p1.x) + (-p2.x + p1.x) * (p.y - p1.y)
}
}
}
```
Output:
```Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0.0, 0.0) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0.0, 1.0) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3.0, 1.0) is within triangle ? false

Triangle[(0.1, 0.1111111111111111), (12.5, 33.333333333333336), (25.0, 11.11111111111111)]
Point (5.414285714285714, 14.349206349206348) is within triangle ? true

Triangle[(0.1, 0.1111111111111111), (12.5, 33.333333333333336), (-12.5, 16.666666666666668)]
Point (5.414285714285714, 14.349206349206348) is within triangle ? true```

## Lua

Translation of: C++
```EPS = 0.001
EPS_SQUARE = EPS * EPS

function side(x1, y1, x2, y2, x, y)
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1)
end

function naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
local checkSide1 = side(x1, y1, x2, y2, x, y) >= 0
local checkSide2 = side(x2, y2, x3, y3, x, y) >= 0
local checkSide3 = side(x3, y3, x1, y1, x, y) >= 0
return checkSide1 and checkSide2 and checkSide3
end

function pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)
local xMin = math.min(x1, x2, x3) - EPS
local xMax = math.max(x1, x2, x3) + EPS
local yMin = math.min(y1, y2, y3) - EPS
local yMax = math.max(y1, y2, y3) + EPS
return not (x < xMin or xMax < x or y < yMin or yMax < y)
end

function distanceSquarePointToSegment(x1, y1, x2, y2, x, y)
local p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1)
local dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength
if dotProduct < 0 then
return (x - x1) * (x - x1) + (y - y1) * (y - y1)
end
if dotProduct <= 1 then
local p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y)
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength
end
return (x - x2) * (x - x2) + (y - y2) * (y - y2)
end

function accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
if not pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y) then
return false
end
if naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y) then
return true
end
if distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE then
return true
end
if distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE then
return true
end
if distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE then
return true
end
return false
end

function printPoint(x, y)
io.write('('..x..", "..y..')')
end

function printTriangle(x1, y1, x2, y2, x3, y3)
io.write("Triangle is [")
printPoint(x1, y1)
io.write(", ")
printPoint(x2, y2)
io.write(", ")
printPoint(x3, y3)
print("]")
end

function test(x1, y1, x2, y2, x3, y3, x, y)
printTriangle(x1, y1, x2, y2, x3, y3)
io.write("Point ")
printPoint(x, y)
print(" is within triangle? " .. tostring(accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)))
end

test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 0)
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 1)
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 3, 1)
print()

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, 25, 11.11111111111111, 5.414285714285714, 14.349206349206348)
print()

test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, -12.5, 16.666666666666668, 5.414285714285714, 14.349206349206348)
print()
```
Output:
```Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 0) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 1) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3, 1) is within triangle? false

Triangle is [(0.1, 0.11111111111111), (12.5, 33.333333333333), (25, 11.111111111111)]
Point (5.4142857142857, 14.349206349206) is within triangle? true

Triangle is [(0.1, 0.11111111111111), (12.5, 33.333333333333), (-12.5, 16.666666666667)]
Point (5.4142857142857, 14.349206349206) is within triangle? true```

## Mathematica/Wolfram Language

```RegionMember[Polygon[{{1, 2}, {3, 1}, {2, 4}}], {2, 2}]
```
Output:
`True`

## Nim

Translation of: Kotlin
```import strformat

const
Eps = 0.001
Eps2 = Eps * Eps

type
Point = tuple[x, y: float]
Triangle = object
p1, p2, p3: Point

func initTriangle(p1, p2, p3: Point): Triangle =
Triangle(p1: p1, p2: p2, p3: p3)

func side(p1, p2, p: Point): float =
(p2.y - p1.y) * (p.x - p1.x) + (-p2.x + p1.x) * (p.y - p1.y)

func distanceSquarePointToSegment(p1, p2, p: Point): float =
let p1P2SquareLength = (p2.x - p1.x) * (p2.x - p1.x) + (p2.y - p1.y) * (p2.y - p1.y)
let dotProduct = ((p.x - p1.x) * (p2.x - p1.x) + (p.y - p1.y) * (p2.y - p1.y)) / p1P2SquareLength
if dotProduct < 0:
return (p.x - p1.x) * (p.x - p1.x) + (p.y - p1.y) * (p.y - p1.y)
if dotProduct <= 1:
let pP1SquareLength = (p1.x - p.x) * (p1.x - p.x) + (p1.y - p.y) * (p1.y - p.y)
return pP1SquareLength - dotProduct * dotProduct * p1P2SquareLength
result = (p.x - p2.x) * (p.x - p2.x) + (p.y - p2.y) * (p.y - p2.y)

func pointInTriangleBoundingBox(t: Triangle; p: Point): bool =
let xMin = min(t.p1.x, min(t.p2.x, t.p3.x)) - EPS
let xMax = max(t.p1.x, max(t.p2.x, t.p3.x)) + EPS
let yMin = min(t.p1.y, min(t.p2.y, t.p3.y)) - EPS
let yMax = max(t.p1.y, max(t.p2.y, t.p3.y)) + EPS
result = p.x in xMin..xMax and p.y in yMin..yMax

func nativePointInTriangle(t: Triangle; p: Point): bool =
let checkSide1 = side(t.p1, t.p2, p) >= 0
let checkSide2 = side(t.p2, t.p3, p) >= 0
let checkSide3 = side(t.p3, t.p1, p) >= 0
result = checkSide1 and checkSide2 and checkSide3

func accuratePointInTriangle(t: Triangle; p: Point): bool =
if not t.pointInTriangleBoundingBox(p):
return false
if t.nativePointInTriangle(p):
return true
if distanceSquarePointToSegment(t.p1, t.p2, p) <= Eps2 or
distanceSquarePointToSegment(t.p3, t.p1, p) <= Eps2:
return true

func `\$`(p: Point): string = &"({p.x}, {p.y})"

func `\$`(t: Triangle): string = &"Triangle[{t.p1}, {t.p2}, {t.p3}]"

func contains(t: Triangle; p: Point): bool = t.accuratePointInTriangle(p)

when isMainModule:

proc test(t: Triangle; p: Point) =
echo t
echo &"Point {p} is within triangle ? {p in t}"

var p1: Point = (1.5, 2.4)
var p2: Point = (5.1, -3.1)
var p3: Point = (-3.8, 1.2)
var tri = initTriangle(p1, p2, p3)
test(tri, (0.0, 0.0))
test(tri, (0.0, 1.0))
test(tri, (3.0, 1.0))
echo()
p1 = (1 / 10, 1 / 9)
p2 = (100 / 8, 100 / 3)
p3 = (100 / 4, 100 / 9)
tri = initTriangle(p1, p2, p3)
let pt = (p1.x + 3.0 / 7 * (p2.x - p1.x), p1.y + 3.0 / 7 * (p2.y - p1.y))
test(tri, pt)
echo()
p3 = (-100 / 8, 100 / 6)
tri = initTriangle(p1, p2, p3)
test(tri, pt)
```
Output:
```Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0.0, 0.0) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0.0, 1.0) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3.0, 1.0) is within triangle ? false

Triangle[(0.1, 0.1111111111111111), (12.5, 33.33333333333334), (25.0, 11.11111111111111)]
Point (5.414285714285714, 14.34920634920635) is within triangle ? true

Triangle[(0.1, 0.1111111111111111), (12.5, 33.33333333333334), (-12.5, 16.66666666666667)]
Point (5.414285714285714, 14.34920634920635) is within triangle ? true```

## Perl

Translate the Java program at this blog post and data set is taken from the Raku entry.

```# 20201123 added Perl programming solution

use strict;
use warnings;

use List::AllUtils qw(min max natatime);
use constant EPSILON        =>           0.001;
use constant EPSILON_SQUARE => EPSILON*EPSILON;

sub side {
my (\$x1, \$y1, \$x2, \$y2, \$x, \$y) = @_;
return (\$y2 - \$y1)*(\$x - \$x1) + (-\$x2 + \$x1)*(\$y - \$y1);
}

sub naivePointInTriangle {
my (\$x1, \$y1, \$x2, \$y2, \$x3, \$y3, \$x, \$y) = @_;
my \$checkSide1 = side(\$x1, \$y1, \$x2, \$y2, \$x, \$y) >= 0 ;
my \$checkSide2 = side(\$x2, \$y2, \$x3, \$y3, \$x, \$y) >= 0 ;
my \$checkSide3 = side(\$x3, \$y3, \$x1, \$y1, \$x, \$y) >= 0 ;
return \$checkSide1 && \$checkSide2 && \$checkSide3  || 0 ;
}

sub pointInTriangleBoundingBox {
my (\$x1, \$y1, \$x2, \$y2, \$x3, \$y3, \$x, \$y) = @_;
my \$xMin = min(\$x1, min(\$x2, \$x3)) - EPSILON;
my \$xMax = max(\$x1, max(\$x2, \$x3)) + EPSILON;
my \$yMin = min(\$y1, min(\$y2, \$y3)) - EPSILON;
my \$yMax = max(\$y1, max(\$y2, \$y3)) + EPSILON;
( \$x < \$xMin || \$xMax < \$x || \$y < \$yMin || \$yMax < \$y ) ? 0 : 1
}

sub distanceSquarePointToSegment {
my (\$x1, \$y1, \$x2, \$y2, \$x, \$y) = @_;
my \$p1_p2_squareLength = (\$x2 - \$x1)**2 + (\$y2 - \$y1)**2;
my \$dotProduct = (\$x-\$x1)*(\$x2-\$x1)+(\$y-\$y1)*(\$y2-\$y1) ;
if ( \$dotProduct < 0 ) {
return (\$x - \$x1)**2 + (\$y - \$y1)**2;
} elsif ( \$dotProduct <= \$p1_p2_squareLength ) {
my \$p_p1_squareLength = (\$x1 - \$x)**2 + (\$y1 - \$y)**2;
return \$p_p1_squareLength - \$dotProduct**2 / \$p1_p2_squareLength;
} else {
return (\$x - \$x2)**2 + (\$y - \$y2)**2;
}
}

sub accuratePointInTriangle {
my (\$x1, \$y1, \$x2, \$y2, \$x3, \$y3, \$x, \$y) = @_;
return 0 unless pointInTriangleBoundingBox(\$x1,\$y1,\$x2,\$y2,\$x3,\$y3,\$x,\$y);
return 1 if ( naivePointInTriangle(\$x1, \$y1, \$x2, \$y2, \$x3, \$y3, \$x, \$y)
or distanceSquarePointToSegment(\$x1, \$y1, \$x2, \$y2, \$x, \$y) <= EPSILON_SQUARE
or distanceSquarePointToSegment(\$x2, \$y2, \$x3, \$y3, \$x, \$y) <= EPSILON_SQUARE
or distanceSquarePointToSegment(\$x3, \$y3, \$x1, \$y1, \$x, \$y) <= EPSILON_SQUARE);
return 0
}

my @DATA = (1.5, 2.4, 5.1, -3.1, -3.8, 0.5);

for my \$point ( [0,0] , [0,1] ,[3,1] ) {
print "Point (", join(',',@\$point), ") is within triangle ";
my \$iter = natatime 2, @DATA;
while ( my @vertex = \$iter->()) { print '(',join(',',@vertex),') ' }
print ': ',naivePointInTriangle (@DATA, @\$point) ? 'True' : 'False', "\n" ;
}
```
Output:
```Point (0,0) is within triangle (1.5,2.4) (5.1,-3.1) (-3.8,0.5) : True
Point (0,1) is within triangle (1.5,2.4) (5.1,-3.1) (-3.8,0.5) : True
Point (3,1) is within triangle (1.5,2.4) (5.1,-3.1) (-3.8,0.5) : False```

## Phix

Both the following as well as some further experiments can be found in demo\rosetta\Within_triangle.exw

### using convex_hull

Using convex_hull() from Convex_hull#Phix

```with javascript_semantics
constant p0 = {0,0},
p1 = {0,1},
p2 = {3,1},
triangle = {{3/2, 12/5}, {51/10, -31/10}, {-19/5, 1/2}}
function inside(sequence p)
return sort(convex_hull({p}&triangle))==sort(deep_copy(triangle))
end function
printf(1,"Point %v is with triangle %v?:%t\n",{p0,triangle,inside(p0)})
printf(1,"Point %v is with triangle %v?:%t\n",{p1,triangle,inside(p1)})
printf(1,"Point %v is with triangle %v?:%t\n",{p2,triangle,inside(p2)})
```
Output:
```Point {0,0} is with triangle {{1.5,2.4},{5.1,-3.1},{-3.8,0.5}}?:true
Point {0,1} is with triangle {{1.5,2.4},{5.1,-3.1},{-3.8,0.5}}?:true
Point {3,1} is with triangle {{1.5,2.4},{5.1,-3.1},{-3.8,0.5}}?:false
```

### trans python

(same output)

```with javascript_semantics
constant p0 = {0,0},
p1 = {0,1},
p2 = {3,1},
triangle = {{3/2, 12/5}, {51/10, -31/10}, {-19/5, 1/2}}

function side(sequence p1, p2, p3)
-- which side of plane cut by line (p2, p3) is p1 on?
atom {x1, y1} = p1,
{x2, y2} = p2,
{x3, y3} = p3
return (x1 - x3) * (y2 - y3) - (x2 - x3) * (y1 - y3)
end function

function iswithin(sequence point, triangle)
--
--  Determine if point is within triangle.
--  If so, the point will be on the same side of each of the half planes
--  defined by vectors p1p2, p2p3, and p3p1. side is positive if outside,
--  negative if inside such a plane. All should be non-negative or all
--  non-positive if the point is within the triangle.
--
sequence {pt1, pt2, pt3} = triangle
atom side12 = side(point, pt1, pt2),
side23 = side(point, pt2, pt3),
side31 = side(point, pt3, pt1)
bool all_non_neg = side12 >= 0 and side23 >= 0 and side31 >= 0,
all_non_pos = side12 <= 0 and side23 <= 0 and side31 <= 0
return all_non_neg or all_non_pos
end function

printf(1,"point %v is with triangle %v?:%t\n",{p0,triangle,iswithin(p0,triangle)})
printf(1,"point %v is with triangle %v?:%t\n",{p1,triangle,iswithin(p1,triangle)})
printf(1,"point %v is with triangle %v?:%t\n",{p2,triangle,iswithin(p2,triangle)})
```

## Python

```""" find if point is in a triangle """

from sympy.geometry import Point, Triangle

def sign(pt1, pt2, pt3):
""" which side of plane cut by line (pt2, pt3) is pt1 on? """
return (pt1.x - pt3.x) * (pt2.y - pt3.y) - (pt2.x - pt3.x) * (pt1.y - pt3.y)

def iswithin(point, pt1, pt2, pt3):
"""
Determine if point is within triangle formed by points p1, p2, p3.
If so, the point will be on the same side of each of the half planes
defined by vectors p1p2, p2p3, and p3p1. zval is positive if outside,
negative if inside such a plane. All should be positive or all negative
if point is within the triangle.
"""
zval1 = sign(point, pt1, pt2)
zval2 = sign(point, pt2, pt3)
zval3 = sign(point, pt3, pt1)
notanyneg = zval1 >= 0 and zval2 >= 0 and zval3 >= 0
notanypos = zval1 <= 0 and zval2 <= 0 and zval3 <= 0
return notanyneg or notanypos

if __name__ == "__main__":
POINTS = [Point(0, 0)]
TRI = Triangle(Point(1.5, 2.4), Point(5.1, -3.1), Point(-3.8, 0.5))
for pnt in POINTS:
a, b, c = TRI.vertices
isornot = "is" if iswithin(pnt, a, b, c) else "is not"
print("Point", pnt, isornot, "within the triangle", TRI)
```
Output:
```Point Point2D(0, 0) is within the triangle Triangle(Point2D(3/2, 12/5), Point2D(51/10, -31/10), Point2D(-19/5, 1/2))
```

## Racket

Racket has exact numbers in its numerical tower... so I don't see much motivation to accomodate rounding errors. This is why the implementation _fails_ the second imprecise test, whereas other implementations pass it. That point is very close to the edge of the triange. If your edge is fat enough (epsilon), it will fall inside. If it is infinitessimal (i.e. exact), it is on the outside.

I would probably use the dot-product version, if only because it requires less (no) division.

```#lang racket/base

(define-syntax-rule (all-between-0..1? x ...)
(and (<= 0 x 1) ...))

(define (point-in-triangle?/barycentric x1 y1 x2 y2 x3 y3)
(let* ((y2-y3 (- y2 y3))
(x1-x3 (- x1 x3))
(x3-x2 (- x3 x2))
(y1-y3 (- y1 y3))
(d (+ (* y2-y3 x1-x3) (* x3-x2 y1-y3))))
(λ (x y)
(define a (/ (+ (* x3-x2 (- y y3)) (* y2-y3 (- x x3))) d))
(define b (/ (- (* x1-x3 (- y y3)) (* y1-y3 (- x x3))) d))
(define c (- 1 a b))
(all-between-0..1? a b c))))

(define (point-in-triangle?/parametric x1 y1 x2 y2 x3 y3)
(let ((dp (+ (* x1 (- y2 y3)) (* y1 (- x3 x2)) (* x2 y3) (- (* y2 x3)))))
(λ (x y)
(define t1 (/ (+ (* x (- y3 y1)) (* y (- x1 x3)) (- (* x1 y3)) (* y1 x3)) dp))
(define t2 (/ (+ (* x (- y2 y1)) (* y (- x1 x2)) (- (* x1 y2)) (* y1 x2)) (- dp)))
(all-between-0..1? t1 t2 (+ t1 t2)))))

(define (point-in-triangle?/dot-product X1 Y1 X2 Y2 X3 Y3)
(λ (x y)
(define (check-side x1 y1 x2 y2)
(>= (+ (* (- y2 y1) (- x x1)) (* (- x1 x2) (- y y1))) 0))
(and
(check-side X1 Y1 X2 Y2)
(check-side X2 Y2 X3 Y3)
(check-side X3 Y3 X1 Y1))))

(module+ main
(require rackunit)

(define (run-tests point-in-triangle?)
(define pit?-1 (point-in-triangle? #e1.5 #e2.4 #e5.1 #e-3.1 #e-3.8 #e1.2))
(check-true (pit?-1 0 0))
(check-true (pit?-1 0 1))
(check-false (pit?-1 3 1))
(check-true ((point-in-triangle? 1/10 1/9 25/2 100/3  25   10/9) #e5.414285714285714 #e14.349206349206348))
; exactly speaking, point is _not_ in the triangle
(check-false ((point-in-triangle? 1/10 1/9 25/2 100/3 -25/2 50/3) #e5.414285714285714 #e14.349206349206348)))

(run-tests point-in-triangle?/barycentric)
(run-tests point-in-triangle?/parametric)
(run-tests point-in-triangle?/dot-product))
```
Output:

no output means all tests passed

## Raku

Reusing code from the Convex hull task and some logic from the Determine if two triangles overlap task.

```class Point {
has Real \$.x is rw;
has Real \$.y is rw;
method gist { [~] '(', self.x,', ', self.y, ')' };
}

sub sign (Point \$a, Point \$b, Point \$c) {
(\$b.x - \$a.x)*(\$c.y - \$a.y) - (\$b.y - \$a.y)*(\$c.x - \$a.x);
}

sub triangle (*@points where *.elems == 6) {
@points.batch(2).map: { Point.new(:x(.[0]),:y(.[1])) };
}

sub is-within (\$point, @triangle is copy) {
my @signs = sign(\$point, |(@triangle.=rotate)[0,1]) xx 3;
so (all(@signs) >= 0) or so(all(@signs) <= 0);
}

my @triangle = triangle((1.5, 2.4), (5.1, -3.1), (-3.8, 0.5));

for Point.new(:x(0),:y(0)),
Point.new(:x(0),:y(1)),
Point.new(:x(3),:y(1))
-> \$point {
say "Point {\$point.gist} is within triangle {join ', ', @triangle».gist}: ",
\$point.&is-within: @triangle
}
```
Output:
```Point (0, 0) is within triangle (1.5, 2.4), (5.1, -3.1), (-3.8, 0.5): True
Point (0, 1) is within triangle (1.5, 2.4), (5.1, -3.1), (-3.8, 0.5): True
Point (3, 1) is within triangle (1.5, 2.4), (5.1, -3.1), (-3.8, 0.5): False```

## REXX

Translation of: Python

Extra certification code was added to verify that the   X,Y   coördinates for the points are not missing and are numeric.

```/*REXX program determines if a   specified point   is within a   specified triangle.    */
parse arg p a b c .                              /*obtain optional arguments from the CL*/
if p=='' | p==","  then p= '(0,0)'               /*Not specified?  Then use the default.*/
if a=='' | a==","  then a= '(1.5,2.4)'           /* "      "         "   "   "     "    */
if b=='' | b==","  then b= '(5.1,-3.1)'          /* "      "         "   "   "     "    */
if c=='' | c==","  then c= '(-3.8,0.5)'          /* "      "         "   "   "     "    */
if  ?(p, a, b, c)  then @= ' is '                /*Is the point  inside the triangle ?  */
else @= " isn't "             /* "  "    "   outside  "      "       */
say 'point'   p    @    " within the triangle "      a     ','       b       ","       c
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
cert: parse arg z,W; if datatype(z,'N')  then return z; call serr z /*return coördinate.*/
serr: say W 'data point '  z  " isn't numeric or missing."; exit 13 /*tell error message*/
x: procedure; parse arg "(" x ','      ;  return cert(x,"X")  /*return the X coördinate.*/
y: procedure; parse arg       ',' y ")";  return cert(y,"Y")  /*   "    "  Y      "     */
\$: parse arg aa,bb,cc;  return (x(aa)-x(cc)) *(y(bb)-y(cc)) - (x(bb)-x(cc)) *(y(aa)-y(cc))
?: #1=\$(p,a,b); #2=\$(p,b,c); #3=\$(p,c,a); return (#1>=0&#2>=0&#3>=0) | (#1<=0&#2<=0&#3<=0)
```
output   when using the default triangle and the point at:   (0,0)
```point (0,0)   is   within the triangle  (1.5,2.4) , (5.1,-3.1) , (-3.8,0.5)
```
output   when using the default triangle and the point at:   (0,1)
```point (0,1)   is   within the triangle  (1.5,2.4) , (5.1,-3.1) , (-3.8,0.5)
```
output   when using the default triangle and the point at:   (3,1)
```point (3,1)   isn't   within the triangle  (1.5,2.4) , (5.1,-3.1) , (-3.8,0.5)
```

## Ruby

Translation of: Go
```EPS = 0.001
EPS_SQUARE = EPS * EPS

def side(x1, y1, x2, y2, x, y)
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1)
end

def naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
checkSide1 = side(x1, y1, x2, y2, x, y) >= 0
checkSide2 = side(x2, y2, x3, y3, x, y) >= 0
checkSide3 = side(x3, y3, x1, y1, x, y) >= 0
return checkSide1 && checkSide2 && checkSide3
end

def pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)
xMin = [x1, x2, x3].min - EPS
xMax = [x1, x2, x3].max + EPS
yMin = [y1, y2, y3].min - EPS
yMax = [y1, y2, y3].max + EPS
return !(x < xMin || xMax < x || y < yMin || yMax < y)
end

def distanceSquarePointToSegment(x1, y1, x2, y2, x, y)
p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1)
dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength
if dotProduct < 0 then
return (x - x1) * (x - x1) + (y - y1) * (y - y1)
end
if dotProduct <= 1 then
p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y)
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength
end
return (x - x2) * (x - x2) + (y - y2) * (y - y2)
end

def accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
if !pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y) then
return false
end
if naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y) then
return true
end
if distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE then
return true
end
if distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE then
return true
end
if distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE then
return true
end
return false
end

def main
pts = [[0, 0], [0, 1], [3, 1]]
tri = [[1.5, 2.4], [5.1, -3.1], [-3.8, 1.2]]
print "Triangle is ", tri, "\n"
x1, y1 = tri[0][0], tri[0][1]
x2, y2 = tri[1][0], tri[1][1]
x3, y3 = tri[2][0], tri[2][1]
for pt in pts
x, y = pt[0], pt[1]
within = accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
print "Point ", pt, " is within triangle? ", within, "\n"
end
print "\n"

tri = [[0.1, 1.0 / 9.0], [12.5, 100.0 / 3.0], [25.0, 100.0 / 9.0]]
print "Triangle is ", tri, "\n"
x1, y1 = tri[0][0], tri[0][1]
x2, y2 = tri[1][0], tri[1][1]
x3, y3 = tri[2][0], tri[2][1]
x = x1 + (3.0 / 7.0) * (x2 - x1)
y = y1 + (3.0 / 7.0) * (y2 - y1)
pt = [x, y]
within = accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
print "Point ", pt, " is within triangle? ", within, "\n"
print "\n"

tri = [[0.1, 1.0 / 9.0], [12.5, 100.0 / 3.0], [-12.5, 100.0 / 6.0]]
print "Triangle is ", tri, "\n"
x3, y3 = tri[2][0], tri[2][1]
within = accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
print "Point ", pt, " is within triangle? ", within, "\n"
end

main()
```
Output:
```Triangle is [[1.5, 2.4], [5.1, -3.1], [-3.8, 1.2]]
Point [0, 0] is within triangle? true
Point [0, 1] is within triangle? true
Point [3, 1] is within triangle? false

Triangle is [[0.1, 0.1111111111111111], [12.5, 33.333333333333336], [25.0, 11.11111111111111]]
Point [5.414285714285714, 14.349206349206348] is within triangle? true

Triangle is [[0.1, 0.1111111111111111], [12.5, 33.333333333333336], [-12.5, 16.666666666666668]]
Point [5.414285714285714, 14.349206349206348] is within triangle? true```

## V (Vlang)

Translation of: go
```import math

const eps = 0.001
const eps_square = eps * eps

fn side(x1 f64, y1 f64, x2 f64, y2 f64, x f64, y f64) f64 {
return (y2-y1)*(x-x1) + (-x2+x1)*(y-y1)
}

fn native_point_in_triangle(x1 f64, y1 f64, x2 f64, y2 f64, x3 f64, y3 f64, x f64, y f64) bool {
check_side1 := side(x1, y1, x2, y2, x, y) >= 0
check_side2 := side(x2, y2, x3, y3, x, y) >= 0
check_side3 := side(x3, y3, x1, y1, x, y) >= 0
return check_side1 && check_side2 && check_side3
}

fn point_in_triangle_bounding_box(x1 f64, y1 f64, x2 f64, y2 f64, x3 f64, y3 f64, x f64, y f64) bool {
x_min := math.min(x1, math.min(x2, x3)) - eps
x_max := math.max(x1, math.max(x2, x3)) + eps
y_min := math.min(y1, math.min(y2, y3)) - eps
y_max := math.max(y1, math.max(y2, y3)) + eps
return !(x < x_min || x_max < x || y < y_min || y_max < y)
}

fn distance_square_point_to_segment(x1 f64, y1 f64, x2 f64, y2 f64, x f64, y f64) f64 {
pq_p2_square_length := (x2-x1)*(x2-x1) + (y2-y1)*(y2-y1)
dot_product := ((x-x1)*(x2-x1) + (y-y1)*(y2-y1)) / pq_p2_square_length
if dot_product < 0 {
return (x-x1)*(x-x1) + (y-y1)*(y-y1)
} else if dot_product <= 1 {
p_p1_square_length := (x1-x)*(x1-x) + (y1-y)*(y1-y)
return p_p1_square_length - dot_product*dot_product*pq_p2_square_length
} else {
return (x-x2)*(x-x2) + (y-y2)*(y-y2)
}
}

fn accurate_point_in_triangle(x1 f64, y1 f64, x2 f64, y2 f64, x3 f64, y3 f64, x f64, y f64) bool {
if !point_in_triangle_bounding_box(x1, y1, x2, y2, x3, y3, x, y) {
return false
}
if native_point_in_triangle(x1, y1, x2, y2, x3, y3, x, y) {
return true
}
if distance_square_point_to_segment(x1, y1, x2, y2, x, y) <= eps_square {
return true
}
if distance_square_point_to_segment(x2, y2, x3, y3, x, y) <= eps_square {
return true
}
if distance_square_point_to_segment(x3, y3, x1, y1, x, y) <= eps_square {
return true
}
return false
}

fn main() {
pts := [[f64(0), 0], [f64(0), 1], [f64(3), 1]]
mut tri := [[3.0 / 2, 12.0 / 5], [51.0 / 10, -31.0 / 10], [-19.0 / 5, 1.2]]
println("Triangle is \$tri")
mut x1, mut y1 := tri[0][0], tri[0][1]
mut x2, mut y2 := tri[1][0], tri[1][1]
mut x3, mut y3 := tri[2][0], tri[2][1]
for pt in pts {
x, y := pt[0], pt[1]
within := accurate_point_in_triangle(x1, y1, x2, y2, x3, y3, x, y)
println("Point \$pt is within triangle? \$within")
}
println('')
tri = [[1.0 / 10, 1.0 / 9], [100.0 / 8, 100.0 / 3], [100.0 / 4, 100.0 / 9]]
println("Triangle is \$tri")
x1, y1 = tri[0][0], tri[0][1]
x2, y2 = tri[1][0], tri[1][1]
x3, y3 = tri[2][0], tri[2][1]
x := x1 + (3.0/7)*(x2-x1)
y := y1 + (3.0/7)*(y2-y1)
pt := [x, y]
mut within := accurate_point_in_triangle(x1, y1, x2, y2, x3, y3, x, y)
println("Point \$pt is within triangle ? \$within")
println('')
tri = [[1.0 / 10, 1.0 / 9], [100.0 / 8, 100.0 / 3], [-100.0 / 8, 100.0 / 6]]
println("Triangle is \$tri")
x3 = tri[2][0]
y3 = tri[2][1]
within = accurate_point_in_triangle(x1, y1, x2, y2, x3, y3, x, y)
println("Point \$pt is within triangle ? \$within")
}```
Output:
```Triangle is [[1.5, 2.4], [5.1, -3.1], [-3.8, 1.2]]
Point [0, 0] is within triangle? true
Point [0, 1] is within triangle? true
Point [3, 1] is within triangle? false

Triangle is [[0.1, 0.1111111111111111], [12.5, 33.333333333333336], [25, 11.11111111111111]]
Point [5.414285714285714, 14.349206349206348] is within triangle ? true

Triangle is [[0.1, 0.1111111111111111], [12.5, 33.333333333333336], [-12.5, 16.666666666666668]]
Point [5.414285714285714, 14.349206349206348] is within triangle ? true
```

## Wren

This is a translation of the ActionScript code for the 'accurate' method in the first referenced article above.

```var EPS = 0.001
var EPS_SQUARE = EPS * EPS

var side = Fn.new { |x1, y1, x2, y2, x, y|
return (y2 - y1)*(x - x1) + (-x2 + x1)*(y - y1)
}

var naivePointInTriangle = Fn.new { |x1, y1, x2, y2, x3, y3, x, y|
var checkSide1 = side.call(x1, y1, x2, y2, x, y) >= 0
var checkSide2 = side.call(x2, y2, x3, y3, x, y) >= 0
var checkSide3 = side.call(x3, y3, x1, y1, x, y) >= 0
return checkSide1 && checkSide2 && checkSide3
}

var pointInTriangleBoundingBox = Fn.new { |x1, y1, x2, y2, x3, y3, x, y|
var xMin = x1.min(x2.min(x3)) - EPS
var xMax = x1.max(x2.max(x3)) + EPS
var yMin = y1.min(y2.min(y3)) - EPS
var yMax = y1.max(y2.max(y3)) + EPS
return !(x < xMin || xMax < x || y < yMin || yMax < y)
}

var distanceSquarePointToSegment = Fn.new { |x1, y1, x2, y2, x, y|
var p1_p2_squareLength = (x2 - x1)*(x2 - x1) + (y2 - y1)*(y2 - y1)
var dotProduct = ((x - x1)*(x2 - x1) + (y - y1)*(y2 - y1)) / p1_p2_squareLength
if (dotProduct < 0) {
return (x - x1)*(x - x1) + (y - y1)*(y - y1)
} else if (dotProduct <= 1) {
var p_p1_squareLength = (x1 - x)*(x1 - x) + (y1 - y)*(y1 - y)
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength
} else {
return (x - x2)*(x - x2) + (y - y2)*(y - y2)
}
}

var accuratePointInTriangle = Fn.new { |x1, y1, x2, y2, x3, y3, x, y|
if (!pointInTriangleBoundingBox.call(x1, y1, x2, y2, x3, y3, x, y)) return false
if (naivePointInTriangle.call(x1, y1, x2, y2, x3, y3, x, y)) return true
if (distanceSquarePointToSegment.call(x1, y1, x2, y2, x, y) <= EPS_SQUARE) return true
if (distanceSquarePointToSegment.call(x2, y2, x3, y3, x, y) <= EPS_SQUARE) return true
if (distanceSquarePointToSegment.call(x3, y3, x1, y1, x, y) <= EPS_SQUARE) return true
return false
}

var pts = [ [0, 0], [0, 1], [3, 1]]
var tri = [ [3/2, 12/5], [51/10, -31/10], [-19/5, 1.2] ]
System.print("Triangle is %(tri)")
var x1 = tri[0][0]
var y1 = tri[0][1]
var x2 = tri[1][0]
var y2 = tri[1][1]
var x3 = tri[2][0]
var y3 = tri[2][1]

for (pt in pts) {
var x = pt[0]
var y = pt[1]
var within = accuratePointInTriangle.call(x1, y1, x2, y2, x3, y3, x, y)
System.print("Point %(pt) is within triangle ? %(within)")
}
System.print()
tri = [ [1/10, 1/9], [100/8, 100/3], [100/4, 100/9] ]
System.print("Triangle is %(tri)")
x1 = tri[0][0]
y1 = tri[0][1]
x2 = tri[1][0]
y2 = tri[1][1]
x3 = tri[2][0]
y3 = tri[2][1]
var x = x1 + (3/7)*(x2 - x1)
var y = y1 + (3/7)*(y2 - y1)
var pt = [x, y]
var within = accuratePointInTriangle.call(x1, y1, x2, y2, x3, y3, x, y)
System.print("Point %(pt) is within triangle ? %(within)")
System.print()
tri = [ [1/10, 1/9], [100/8, 100/3], [-100/8, 100/6] ]
System.print("Triangle is %(tri)")
x3 = tri[2][0]
y3 = tri[2][1]
within = accuratePointInTriangle.call(x1, y1, x2, y2, x3, y3, x, y)
System.print("Point %(pt) is within triangle ? %(within)")
```
Output:
```Triangle is [[1.5, 2.4], [5.1, -3.1], [-3.8, 1.2]]
Point [0, 0] is within triangle ? true
Point [0, 1] is within triangle ? true
Point [3, 1] is within triangle ? false

Triangle is [[0.1, 0.11111111111111], [12.5, 33.333333333333], [25, 11.111111111111]]
Point [5.4142857142857, 14.349206349206] is within triangle ? true

Triangle is [[0.1, 0.11111111111111], [12.5, 33.333333333333], [-12.5, 16.666666666667]]
Point [5.4142857142857, 14.349206349206] is within triangle ? true
```

## XPL0

```func real Dot(W,X,Y,Z);         \Return the dot product of two 2D vectors
real W,X,Y,Z;                   \ (W-X) dot (Y-Z)
real WX(2), YZ(2);
[WX(0):= W(0)-X(0);  WX(1):= W(1)-X(1);
YZ(0):= Y(0)-Z(0);  YZ(1):= Y(1)-Z(1);
return WX(0)*YZ(0) + WX(1)*YZ(1);
];

real A,B,C;                     \triangle

func PointInTri(P);             \Return 'true' if point P is inside triangle ABC
real P;
int  S0,S1,S2;          \signs
[S0:= Dot(P,A,B,A) >= 0.0;
S1:= Dot(P,B,C,B) >= 0.0;
S2:= Dot(P,C,A,C) >= 0.0;
return S0=S1 & S1=S2 & S2=S0;
];

[A:= [10.5, 6.3];  B:= [13.5, 3.6];  C:= [ 3.3, -1.6];
Text(0, if PointInTri([10.0, 3.0]) then "inside" else "outside");  CrLf(0);
Text(0, if PointInTri([-5.0,-2.2]) then "inside" else "outside");  CrLf(0);
Text(0, if PointInTri([10.5, 6.3]) then "inside" else "outside");  CrLf(0);
]```
Output:
```inside
outside
inside
```