History and Background

Problem. Let $ABC$ be an isosceles triangle (AB=AC) with $\angle BAC = 20^\circ$. Point $D$ is on side $AC$ such that $\angle DBC = 60^\circ$. Point $E$ is on side $AB$ such that $\angle ECB = 50 ^\circ$. Find, with proof, the measure of $\angle EDB$.

This geometry problem which is the major focus of the talk today dates back to at least 1922, when it appeared in the Mathematical Gazette, Volume 11, p. 173. It appears to be an easy problem, but it is deceivingly difficult. I first saw the problem in the late 1960's when my go teacher at the time gave it to me. I worked for a long time on the problem. However, I persisted in treating it as an elementary angle-chasing problem and did not solve it. A few years later the problem surfaced in a talk for high school students by Bill Leonard from California State College in Fullerton. As soon as he showed the problem on the screen, I covered my ears and put my head down, because I did not want to hear the solution. Nevertheless, I did not manage to solve the problem and finally when I encountered it in Geometry Revisited [1] by Coxeter and Greitzer, I read the solution one line at a time, trying to complete the proof. It was still difficult for me, since I was teaching Junior High School at the time and had not studied any geometry since high school. I then found the problem in some books I ordered for my classes: Trigonometric Novelties [10] and One Hundred Mathematical Curiosities [9]. Both were written by William Ransom in the 1950's. I had not even considered trying to prove it using trigonometry. Of course there were no calculators in the early 1970's selling for less than $200 that had trigonometric function keys . The Mathematical Association of America was now publishing a new series of books, the Dolciani Mathematical Expositions. The second volume, Mathematical Gems II [3] by Ross Honsberger, had a section entitled ``Four Minor Gems from Geometry''. In this section, a 1951 proof by S.T. Thompson was presented based on intersecting diagonals of a regular 18-gon. Also during this period, a problem with $60^\circ$ and $70^\circ$ angles instead of $60^\circ$ and $50^\circ$ angles surfaced in the 1976 Carleton University Mathematics Competition for high school students. It was widely discussed in Crux Mathematicorum [2] with a call for non-trigonometric solutions. Many were forthcoming in the months that followed. In the May-June 1994 issue of Quantum there appeared a very interesting article entitled `` Nine Solutions to One Problem'' [5] by Constantine Knop. This talk will be a discussion of these solutions. In 1997, an article in Mathematical Horizons [7] contained an article entitled ``A Better Angle From Outside'' by Andy Liu which discusses several problems that can be solved with the key idea used in the sixth proof in Knop's article. In 2000, Essays on Numbers and Figures [8] by V.V. Prasolov became Volume 16 in the American Mathematical Society series Mathematical World. The essay in this volume, ``Intersection Points of the Diagonals of Regular Polygons'', was to be the main topic of the talk today, but there is too much else to discuss, so it will only be mentioned as a generalization of the method of S.T. Thompson, mentioned above. Last year, Mathematical Chestnuts from Around the World [4] by Ross Honsberger, was published as Dolciani Mathematical Exposition Number 24. A problem from Knop's article, proposed by a gifted ninth grader, is discussed and three solutions are given. (See problem 1 below) Also last year, an article entitled``Dividable Triangles--What Are They?'' came out in the May issue of Mathematics Teacher [6] which approaches these problems from the different point of view of dissecting isosceles triangles into isosceles triangles.