One of the most challenging and rewarding thing associated with being a math enthusiast (a.k.a. mathematician) is an opportunity to share your knowledge about the not so obvious truths of mathematics. A couple of years ago, I tried to communicate that feeling through an article for high school students.

When I joined college, I tried to teach mathematics to some kids from financially not-so strong family. Since they had no exposure to mathematics, I had to start with concepts like addition and multiplication of numbers. My experience can be summarized as the following stand-up comedy performance by Naveen Richard:

After trying for about a couple of months to teach elementary mathematics, I gave up and now I discuss mathematics only above the high school level. Last week I delivered a lecture discussing the proof of Poncelet’s Closure Theorem:

Whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics.

I had spent sufficient time preparing the lecture, and believed that I was aware of all possible consequences of this theorem. But, almost half way through the lecture one person (Haresh) from the audience of 10 people, pointed out following fascinating consequence of the theorem:

If an n-sided polygon is inscribed in one conic section and circumscribed by the other one, then it must be a convex polygon and no other m-sided polygon (with m≠n) can be inscribed and circumscribed by this pair of conic sections.

This kind of insights by audience motivates me to discuss mathematics with others!