Saturday, December 24, 2011

"I think I discovered a theorem, Mr. Karafiol!"

This past week, one of my students stopped me on my first pass around the class with the exciting words "I think I discovered a theorem!  In a right triangle, the angles the median makes with the hypotenuse are twice the other angles!"  We had discussed the median to the hypotenuse in the previous class, but I'd never thought about the angles before.  So I started doing algebra in my head, confirming the relationship he was describing:

Before I could get very far, Vik showed me the elegant diagram below:
With the inscribed angle theorem, the proof is immediate.

Eureka!

2 comments:

  1. I tried to confirm the relationship without looking at the second diagram, and I came up with another way to see it - if you draw the bisector to either of the angles created by the median line, you get a triangle that is similar to the original triangle, which proves it as well. Neat!

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  2. THAT is so cool! Will definitely be sharing this with my students this year and giving credit to Vik.

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