## Workshop: Session 4 — Summary

We began by looking at three proofs of one of my favorite theorems in **geometry**: the line segment connecting the midpoints of two sides of a triangle is parallel to, and half the length of, the third side.

**Proof 1** was the “textbook” proof involving the **Side-Angle-Side Similarity Theorem**: prove the small triangle is similar to the large one by SASS. Congruent corresponding angles give you parallel lines and proportions gives you the length.

**Proof 2** is one of the reasons I love this theorem so much: I use it as a transition back to **coordinate geometry**. There is certainly a lot of background work hiding in the shadows (the** distance** and **midpoint** formulas; tying slopes to parallelism and perpendicularity; establishing what **arbitrary figures** really are), but a few simple calculations give you both the length and the parallelism.

Before really using coordinate geometry to **prove** theorems, however, make sure you fully explore the idea of an **arbitrary figure**. Ask students to sketch and label an arbitary quadrilateral in the **xy-plane**: if a square, rectangle, trapezoid, or kite come back, try again!

**Proof 3** is an **elegant** proof, recently shown to me by a student. Just rotate the triangle 180 degrees around the midpoint!

Just convince yourself that M, N, and M’ are all **collinear** and that CABA’ is a **parallelogram**, and you’re basically done!

As an extension of this idea, we discussed Varignon’s Theorem: take any **quadrilateral**, connect the midpoints of adjacent sides and a parallelogram is formed! Even more amazing is that this theorem holds for **concave **and even** complex quadrilaterals!**

This is a great activity for paper-and-pencil exploration, or with dynamic geometry software like Geogebra.

We also talked about a lovely problem from the 2011 AIME that incorporated a similar idea, namely, taking a scalene triangle and bending the corners down to make a table.