I've just found this interesting question on StackExchange:

If you have a rectangle ABCD and point P. Is P inside ABCD?

## The idea

The idea how to solve this problem is simply beautiful.

If the point is in the rectangle, it divides it into four triangles:

If P is not inside of ABCD, you end up with somethink like this:

You might note that the area of the four triangles in is bigger than the area of the rectangle. So if the area is bigger, you know that the point is outside of the rectangle.

## Formulae

If you know the coordinates of the points, you can calculate the area of the rectangle like this:

\(A_\text{rectangle} = \frac{1}{2} \left| (y_{A}-y_{C})\cdot(x_{D}-x_{B}) + (y_{B}-y_{D})\cdot(x_{A}-x_{C})\right|\)

The area of a triangle is: \(A_\text{triangle} = \frac{1}{2} (x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2))\)

## Python

```
def isPinRectangle(r, P):
"""
r: A list of four points, each has a x- and a y- coordinate
P: A point
"""
areaRectangle = 0.5 * abs(
# y_A y_C x_D x_B
(r[0][1] - r[2][1]) * (r[3][0] - r[1][0])
# y_B y_D x_A x_C
+ (r[1][1] - r[3][1]) * (r[0][0] - r[2][0])
)
ABP = 0.5 * (
r[0][0] * (r[1][1] - r[2][1])
+ r[1][0] * (r[2][1] - r[0][1])
+ r[2][0] * (r[0][1] - r[1][1])
)
BCP = 0.5 * (
r[1][0] * (r[2][1] - r[3][1])
+ r[2][0] * (r[3][1] - r[1][1])
+ r[3][0] * (r[1][1] - r[2][1])
)
CDP = 0.5 * (
r[2][0] * (r[3][1] - r[0][1])
+ r[3][0] * (r[0][1] - r[2][1])
+ r[0][0] * (r[2][1] - r[3][1])
)
DAP = 0.5 * (
r[3][0] * (r[0][1] - r[1][1])
+ r[0][0] * (r[1][1] - r[3][1])
+ r[1][0] * (r[3][1] - r[0][1])
)
return areaRectangle == (ABP + BCP + CDP + DAP)
```

## Triangle

The same idea can easily be adopted to triangles:

```
#!/usr/bin/env python
# -*- coding: utf-8 -*-
class Point:
"""Represents a two dimensional point."""
def __init__(self, x, y):
self.x = x
self.y = y
def __get__(self, obj, cls=None):
return obj
def __repr__(self):
return "P(%.2lf|%.2lf)" % (self.x, self.y)
def __str__(self):
return repr(self)
class Triangle:
"""Represents a triangle in R^2."""
epsilon = 0.001
def __init__(self, a, b, c):
assert isinstance(a, Point)
assert isinstance(b, Point)
assert isinstance(c, Point)
self.a = a
self.b = b
self.c = c
def getArea(self):
"""Get area of this triangle.
>>> Triangle(Point(0.,0.), Point(10.,0.), Point(10.,10.)).getArea()
50.0
>>> Triangle(Point(-10.,0.), Point(10.,0.), Point(10.,10.)).getArea()
100.0
"""
a, b, c = self.a, self.b, self.c
return abs(a.x * (b.y - c.y) + b.x * (c.y - a.y) + c.x * (a.y - b.y)) / 2
def isInside(self, p):
"""Check if p is inside this triangle."""
assert isinstance(p, Point)
currentArea = self.getArea()
pab = Triangle(p, self.a, self.b)
pac = Triangle(p, self.a, self.c)
pbc = Triangle(p, self.b, self.c)
newArea = pab.getArea() + pac.getArea() + pbc.getArea()
return abs(currentArea - newArea) < Triangle.epsilon
if __name__ == "__main__":
import doctest
doctest.testmod()
```

## Credits

Thank you Teon Brooks for reporting an error (I wrote "rectangles" instead of "triangles")