When we accidentally derived the law of cosines

We’ve been working through some of the challenging proofs in chapter 9 of Art of Problem Solving’s Introduction to Geometry book. As I’ve written, my son’s been finding these proofs to be pretty difficult:

BUT – he’s staying with them and seeming to get a little more out of each one that we go through.

Today’s challenge was prove that in an obtuse triangle, if we call the sides A, B, and C, with C opposite the obtuse angle, then $C^2 > A^2 + B^2.$

We spent about 30 minutes working through this proof and then I wanted to go through it again on camera. To my surprise my son took the proof in a completely different direction in the video and we ended up essentially proving the law of cosines by accident. Ha!

The first 5 minutes was introducing the problem, drawing a little picture, and starting down the path toward the proof:

In the second half, we looked at how our picture helped is sort through some complicated looking equations. It doesn’t seem to be progressing too quickly, then at 2:23 – “oooooooooohhh”

I wrap up at the end talking very briefly about the interesting expression that we stumbled on – the law of cosines!