Midpoints (3): A return to quadrilaterals
Constanze, a fellow proofer from Distributed Proofreaders, took me up on my request to find a geometric proof that the midpoints of quadrilaterals form a parallelogram:
To prove that M1M2M3M4 is a parallelogram, we need to prove that M1M2 || M4M3 and M1M4 || M2M3 (definition of a parallelogram).
In Euclidian geometry, two lines are parallel if both are parallel to a third one. The sides of the parallelogram are parallel to the diagonals of the quadrilateral. This can be shown by looking at similarities between triangles: I’ll do this for P1, for the other 3 vertices similar arguments can be made.
Triangles P1M1M4 and P1P2P4 are similar because they share an angle M4P1M1 and the adjacent sides to this angle are proportional: P1M4 / P1P4 = P1M1 / P1P2 = 1/2.
From this follows that angle P1M1M4 = angle P1P2P4 (similar triangles have identical angles). Using the theorem on angles on a transversal to two parallels this means that M1M4 || P2P4.
Discussion of the Geometric Proof
First, note that this takes a completely different route to proving the property than the vector proof I gave in my original post. The key to this geometric proof lies in the observation that the edges of the midpoint parallelogram are parallel to the diagonals of the original quadrilateral. Once this has been noticed, the rest of the proof is just chasing definitions.
Second, I find it very hard to grasp a geometric proof without actually drawing a diagram. So here’s a picture, showing the two diagonals of the quadrilateral:
Costanze’s proof concentrates on one of the four edges of the quadrilateral. Here’s a diagram without the extraneous material:
Comparing the Proofs
Both the vector and the geometric proof are equally valid, but both have different characters, and can be easily generalised in different ways. With the vector proof, for example, we could easily start looking at points which are, say, 10% of the way from one vertex to another. It also doesn’t require any extra diagrams. Technically the geometric proof doesn’t require diagrams either, but I would be very surprised if anyone ever does any geometric proof without having some sort of picture in mind.
One advantage of the geometric proof is that, by identifying the importance of the diagonals of the original quadrilateral, it helps us to answer some of the additional questions raised in the original post. For example, for our midpoint parallelogram to be a rectangle, we need the diagonals of the quadrilateral to be perpendicular. This means that the midpoint parallelogram is certainly a rectangle if our quadrilateral is a kite (or inverted kite). Deciding whether this sufficient condition is necessary (and the further question about when the parallelogram is a square) is an exercise for the reader!
Why Bother?
After we have one proof of a proposition, why should we bother looking for other ones? I run in to questions like this one all the time in teaching, particularly when I am trying to get my students to find their own explanations. There are several reasons.
For one, it is a useful mental exercise to think about a situation in different ways (for example, finding the minimum point of a quadratic using differentiation, and also using completing the square).
Secondly, different proofs of the same proposition can differ hugely in elegance (an area which is incredibly hard to discuss at school level, particularly as we seem to have almost completely removed proof from anything lower than Further Mathematics).
Thirdly, as mentioned above, different styles of mathematical thinking lead themselves to being generalised in different ways (we can easily prove things with vectors in more than three dimensions, which is certainly hard to picture!).
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August 5th, 2008 at 10:04 am
Now my brain hurts