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## Even More Serendipity

In the Math News for this week, I revisit Beulah I. Shoesmith, exemplary math teacher. She served as Chairman of the Department of Mathematics at Hyde Park High School (Chicago) for almost four decades before retiring in 1945.

With students, Beulah claimed that the I. for her middle name was for "Isosceles." In his note, F.R. casts some doubt on this middle-name claim, and clears up some of the mystery.

Every term, Beulah presented her Plane Geometry class with a problem to solve. It was a difficult problem, that supposedly required an indirect proof.

Your task: Prove using methods of Plane Geometry: "If a triangle has two angle bisectors of equal length, then the triangle is isosceles."

Want some more serendipity? In the 1980s on the first day in my Modern Geometries class (university-level), I used to present this same problem to my students (but never having heard of Beulah I. shoesmith). To the class, I stated that if anyone could prove the theorem by the end of the term, they would get an automatic A.

Now, this was before the age of the Internet, so no on-line searches were possible. For more than five years, no students posted solutions, though many tried.

Then, in 1989, a clever student handed in a correct proof on the last day of the term. When I announced his success to the class, the student confessed the following story with humor.

The story: While sitting as an observer in a local high school geometry class, he browsed the class text...and discovered the same problem as a "Bonus" problem at the end of a chapter...complete with the hint needed to prove the theorem.

In case you are curious, I had intended to give him an A, but he had already earned it from his other work in class.

NOTE: I no longer give the problem....any one know of another good one I can use?

Hint: As Beulah perhaps would say: "Hang in there...remember, the problem is difficult but do-able."

Solution Commentary: The problem is famous, an known as the Steiner-Lehmus problem. It is considered to be one of the more difficult problems in classical Euclidean geometry. Plus, multiple different proofs exist for it.

Some "solution" resources for you to browse:

• Answers.Yahoo provides some posed solutions, that you will need to wade through
• Wikipedia provides information regarding the Steiner-Lehmus Theorem, its solution, and its history
• Math Pages provides an interesting perspective on when or when not, regarding the desired proof.
• Roger Cooke's paper is a great overview of the problem and its solution...in the manner I expect Beulah would want.