\documentclass[11pt]{scrartcl}
\usepackage{evan}

\begin{document}
\title{Twitch 125.2}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 125}
\maketitle

\section*{Problem}
Regular hexagon $ABCDEF$ is drawn, and a point $P$ inside this hexagon
satisfies $[PAB]=6$, $[PCD]=10$, and $[PEF]=11$.
Compute $[PBC][PDE][PFA]$.

\section*{Video}
\href{https://www.youtube.com/watch?v=3DdyaBnRYPN30&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/3DdyaBnRYPN30}}



\newpage

\section*{Solution}
Let $x$ denote the side length of the hexagon,
so the heights from $P$ to $AB$, $CD$, $EF$ are
$\frac{12}{x}$, $\frac{20}{x}$, $\frac{22}{x}$ respectively.
Let $h_1$, $h_2$, $h_3$ be the heights from $P$ to $BC$, $DE$, $FA$; then
\begin{align*}
  \sqrt3 x &= \frac{12}{x} + h_1 \\
  &= \frac{20}{x} + h_2 \\
  &= \frac{22}{x} + h_3.
\end{align*}
On the other hand, by considering the equilateral triangles
formed by extending the sides of the hexagon (or quoting Vivani theorem),
\[ h_1 + h_2 + h_3 = \frac{12}{x} + \frac{20}{x} + \frac{22}{x}. \]
Hence, it follows that
\[ 3 \cdot \sqrt 3 x
  = 2 \left( \frac{12}{x} + \frac{20}{x} + \frac{22}{x} \right) \]
so $x^2 = 12\sqrt 3$.

To finish up, compute
\begin{align*}
  [PDE] &= \half x \cdot h_1 &= \frac{\sqrt3}{2} x^2 - 6 = 12 \\
  [PFA] &= \half x \cdot h_2 &= \frac{\sqrt3}{2} x^2 - 10 = 8 \\
  [PBC] &= \half x \cdot h_3 &= \frac{\sqrt3}{2} x^2 - 11 = 7
\end{align*}
giving answer $12 \cdot 8 \cdot 7 = 672$.

\end{document}
