Angles of Reflection: "I think I discovered a theorem, Mr. Karafiol!"

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:

ShawDecember 25, 2011 at 10:24 AM
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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Laura KaplanDecember 25, 2011 at 11:15 AM
THAT is so cool! Will definitely be sharing this with my students this year and giving credit to Vik.
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About Us

John Benson taught for forty-two years as a classroom teacher in public schools. The first year was at Frederick Douglass High School in Atlanta, Georgia, and the next forty-one years, at Evanston Township High School in Evanston, Illinois. He has been actively involved in many math organizations and attributes much of his success as a teacher to this participation. He was founder and president of the North Suburban Math league, spent twelve years grading AP Calculus exams, and taught for many summers at the Center for Talent Development at Northwestern University. He was the Presidential Awardee in Mathematics for Illinois in 1987. He has been a contributing author for four secondary math textbooks, most recently the third edition of UCSMP Geometry.

P.J. Karafiol has taught math for 18 years: at Phillips Academy in Andover, Massachusetts; at Providence-St. Mel School on Chicago's West Side; and for the last twelve years, at Walter Payton College Prep High School in Chicago, where he is the Coordinator for Curriculum, Instruction, and Assessment. Like John, he attributes any success he has had to reflection on past failures and to discussions with his friends and colleagues. He is the lead author of the ARML competition and a co-author of many other contests. He was the Presidential Awardee in mathematics in Illinois in 2009. He is the co-author of two textbooks, the third editions of UCSMP Advanced Algebra and Functions, Statistics, and Trigonometry.