An Informal Proof of Lockhart's Inscribed Parallelogram Theorem

Screenshots from this video: https://www.youtube.com/watch?v=V1gT2f3Fe44

Lockhart observes in his book Measurement that if you mark and connect the midpoints of the sides of ANY quadrilateral, you will inscribe a parallelogram. 

Here is my informal proof of why that is:

1. By induction, any quadrilateral can be formed by joining 3 pairs of congruent triangles where all the triangles are similar. 

2. In order to form a quadrilateral from such a set, one and only one pair of congruent triangles must share a side. 

3. The pair that share a side must form a parallelogram because they are congruent.

4. In order for the quadrilateral to have straight sides, all the triangles must be similar. That is, the other four triangles must share a side with the ones forming a parallelogram. 

5. Marking and joining the midpoints must create a set of congruent triangles because each set of opposite midpoints will be equidistant from midpoint of the shared side where they form a parallelogram. 

Not an elegant proof, but hey, at least it's shorter than that time Russell and Whitehead took 360+ pages to prove that 1 + 1 = 2. 

https://en.wikipedia.org/wiki/Principia_Mathematica

As a lemma, if you connect the opposite instead of adjacent midpoints, you will cut the figure into four smaller versions of itself. 

There's probably an interesting progression for how many congruent circles of a given number can fit inside a larger circle. I'll leave that as an exercise for the reader.