Cauchy’s Functional Equation and Zorn’s Lemma

This is a draft of an appendix chapter for my Napkin project. In the world of olympiad math, there's a famous functional equation that goes as follows: $latex \displaystyle f : {\mathbb R} \rightarrow {\mathbb R} \qquad f(x+y) = f(x) + f(y). &fg=000000$ Everyone knows what its solutions are! There's an obvious family of solutions… Continue reading Cauchy’s Functional Equation and Zorn’s Lemma

Three Properties of Isogonal Conjugates

In this post I'll cover three properties of isogonal conjugates which were only recently made known to me. These properties are generalization of some well-known lemmas, such as the incenter/excenter lemma and the nine-point circle. 1. Definitions Let $latex {ABC}&fg=000000$ be a triangle with incenter $latex {I}&fg=000000$, and let $latex {P}&fg=000000$ be any point in… Continue reading Three Properties of Isogonal Conjugates

Why do roots come in conjugate pairs?

This is an expanded version of an answer I gave to a question that came up while I was assisting the 2014-2015 WOOT class. It struck me as an unusually good way to motivate higher math using stuff that people notice in high school but for some reason decide to not think about. In high… Continue reading Why do roots come in conjugate pairs?