I didn’t encounter the Quadrilateral Midpoint Theorem (QMT) until I had been teaching a few years. Following is a minor variation on my approach to the QMT this year plus a fun way I leveraged the result to introduce similarity.
In case you haven’t heard of it, the surprisingly lovely QMT says that if you connect, in order, the midpoints of the four sides of a quadrilateral–any quadrilateral–even if the quadrilateral is concave or if its sides cross–the resulting figure will always be a parallelogram.
SKETCH OF THE TRADITIONAL PROOF: The proof is often established through triangle similarity: Whenever you connect the midpoints of two sides of a triangle, the resulting segment will be parallel to and half the length of the triangle’s third side. Draw either diagonal in the quadrilateral to create two triangles. Connecting the midpoints of the other two sides of each triangle creates two congruent parallel sides, so the quadrilateral connecting all four midpoints must be a parallelogram.
NEW APPROACH THIS YEAR: I hadn’t yet led my class into similarity, but having just introduced coordinate proofs, I tried an approach I’d never used before. I assigned a coordinate proof of the QMT. I knew the traditional approach existed, but I wanted them to practice their new technique. From a lab in December, they already knew the result of the QMT, but they hadn’t proved it.
PART I: Let quadrilateral ABCD be defined by the points , A=(a,b), B=(c,d), C=(e,f), and D=(g,h). There are several ways to prove that the midpoints of ABCD are the vertices of a parallelogram. Provide one such coordinate proof.
All groups quickly established the midpoints of the four sides: , , , and . From there, my students took three approaches to the final proof, each relying on a different sufficiency condition for parallelograms.
The most common was to show that opposite sides were parallel. and , making those two midpoint segments parallel. Likewise, , proving the other opposite side pair also was parallel. With both pairs of opposite sides parallel, the midpoint quadrilateral was necessarily a parallelogram.
I had two groups leverage the fact that the diagonals of parallelograms were mutually bisecting. . QED.
One student even proved that opposite sides were congruent.
While it was not readily available for my students this year, I can imagine allowing CAS for these manipulations if I use this activity in the future.
EXTENDING THE QMT TO SIMILARITY: For the next stage, I asked my students to explains what happens when the QMT is applied to degenerate quadrilaterals.
PART II: You could think of triangles as being degenerate quadrilaterals when two quadrilateral vertices coincide to make one side of the quadrilateral have side length 0. Apply this to generic quadrilateral ABCD from above where points A and D coincide to create triangle BCD. Use this to explain how the segment connecting the midpoints of any two sides of a triangle is related to the third side of the triangle.
I encourage you to construct this using a dynamic geometry package, but here’s the result.
Heres a brief video showing the quadrilateral going degenerate.
Notice the parallelogram still exists and forms two midpoint segments on the triangle (degenerate quadrilateral). By parallelogram properties, each of these segments is parallel and congruent to the opposite side of the parallelogram, making them parallel to and half the length of the opposite side of the triangle.
CONCLUSION: I think it critical to teach in a way that draws connections between ideas and units. This exercise made a lovely transition from quadrilaterals through coordinate proofs to the triangle midpoint theorem.