I Fought Galois and Galois Won
Good Morning! My sophomore year in high school (1991-1992), while I was learning about right triangles, geometric means, and similar triangles, I asked myself what I thought was a fairly simple and obvious question; if I continue to draw the altitudes for the new triangles formed by the original altitude, will I ever get triangles that are congruent in addition to being similar? For those who are not math-savvy, congruence is a stronger form of similarity. The following images illustrate the idea. The first altitude. The second altitude. The nth altitude. My question is about when triangle CBD 1 is congruent to triangle AD n-1 D n . Using some basics of trigonometry, over the years I was able to write a polynomial equation that I could use to answer my question. In fact, I was able to determine that there was a specific right triangle which produced congruent triangles with the first altitude. Then, using the quadratic formula, I was able to determine a specific triangle which produc