Sharing Stewart’s theorem with my son

My older son had a problem about finding the length of an angle bisector in a 3-4-5 triangle in his enrichment math class last week. Solving this problem is a little tedious, but also gives a great opportunity to introduce Stewart’s theorem. I first learned about Stewart’s theorem from Geometry Revisited when I was in high school. Here’s an explanation of the theorem on Wikipedia:

Stewart’s theorem on Wikipedia

I started off the project tonight by reviewing the original problem with my son:

Next I briefly introduced the theorem and then we got interrupted by someone knocking on our front door:

Now I showed how the proof goes. We had a brief discussion / reminder about the relationship between \cos(\theta) and \cos( 180 - \theta ) and after that the proof went pretty quickly:

Finally, we returned to our original triangle to compute the length of the angle bisector using Stewart’s Theorem. The computation is still a little long, but now the calculations themselves are pretty straightforward:

Definitely a beautiful theorem. It is amazing that the law of cosines simplifies so nicely and that computing the lengths of cevians of a triangle.


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