**• **** example**:

Graph with PHP.

With desmos:

**• ** If the polygon is a triangle you can study it with other scripts.
This script warns, however, if the triangle is rectangle:

**• ** I connect the opposite midpoints on the sides of a convex
(i.e. with all angles less than 180°) quadrilateral to form line segments.
These line segments formed 4 quadrilaterals within the original quadrilateral.
Is the sum of the areas of the non-adjacent quadrilateral equal to the sum of the remaining quadrilaterals?

A + B = C + D ? |

I try with the four vertices P1 = (0,0), P2 = (6,1), P3 = (8,5), P4 = (2,6) (see the picture).

The midponts are M1 = (6/2,1/2) = (3,0.5), M2 = (6+(8-6)/2,1+(5-1)/2) = (7,3), M3 = (8+(2-8)/2,5+(6-5)/2) = (5,5.5),
M4 = (2/2,6/2) = (1,3).

I find the intersection of M1-M3 and M2-M4 with line-line:

DBCAEnter x Enter x Enter x Enter x 4, 1, 0, 3 4, 7, 6, 3 4, 5, 8, 7 4, 5, 2, 1 Enter y Enter y Enter y Enter y 3, 3, 0, 0.5 3, 3, 1, 0.5 3, 5.5, 5, 3 3, 5.5, 6, 3area = 8area = 6.5area = 7area = 8.5

D + C = 8 + 7 = 15; A + B = 8.5 + 6.5 = 15. **OK**

I also try with (0,4), (5,0), (13,10), (6,11). Midpoints: (2.5,2), (9,5), (9.5,10.5), (3,7.5). Intersection of M1-M3 and M2-M4: |

x y D 6, 3, 0, 2.5 6.25, 7.5, 4, 2 area = 15.9375 B 6, 2.5, 5, 9 6.25, 2, 0, 5 area = 18.8125 C 6, 9, 13, 9.5 6.25, 5, 10, 10.5 area = 18.3125 A 6, 9.5, 6, 3 6.25, 10.5, 11, 7.5 area = 15.4375

D + C = 15.9375 + 18.3125 = 34.25; A + B = 15.4375 + 18.8125 = 34.25. **OK**