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 arbitrary quadrilateral in the xy-plane:   if a square, rectangle, trapezoid, or kite comes back, try again!

Proof 3 is an elegant proof 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 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.

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