Kirchberger's theorem

Imagine you have a shape with a bunch of sides, corners, and angles. If we draw a line from a corner to the midpoint of the opposite side, and then another line from a different corner to the midpoint of the opposite side, what happens? Well, Kirchberger's Theorem tells us that these two lines will be the same length if and only if the shape that we started with is what we call a parallelogram. This is like saying that if we take two corners of a parallelogram and draw lines to the midpoint of the opposite side, the two lines will be equal in length. But if we try this trick on some other shape, like a triangle or a pentagon, the lines we draw won't be equal in length. So Kirchberger's Theorem helps us understand what a parallelogram is and how it's different from other shapes.