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

\begin{document}
\title{EMCC 2023 T-14}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 138}
\maketitle

\section*{Problem}
Equilateral hexagon $ABCDEF$ has opposite sides parallel,
side length $3$, and area $5$.
Find $AC\cdot BD\cdot CE\cdot DF\cdot EA\cdot FB$.

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



\newpage

\section*{Solution}
Construct parallelogram $COED$.
Then $ABCO$ is also a parallelogram
(because $\vec B - \vec A = \vec D - \vec E = \vec C - \vec O$),
and similarly $AFEO$ is a parallelogram.
So the point $O$ thus the circumcenter of triangle $ACE$,
because $AO = CO = EO = 3$, as shown below.

\begin{center}
\begin{asy}
pair A = dir(110);
pair C = dir(210);
pair E = dir(330);
pair B = A+C;
pair D = C+E;
pair F = A+E;
pair O = origin;

filldraw(A--C--E--cycle, opacity(0.1)+yellow, red);
filldraw(A--B--C--D--E--F--cycle, opacity(0.1)+palecyan, blue);
draw(O--A, deepgreen);
draw(O--C, deepgreen);
draw(O--E, deepgreen);

dot("$A$", A, dir(A));
dot("$C$", C, dir(C));
dot("$E$", E, dir(E));
dot("$B$", B, dir(B));
dot("$D$", D, dir(D));
dot("$F$", F, dir(F));
dot("$O$", O, dir(270));

/* -----------------------------------------------------------------+
|                 TSQX: by CJ Quines and Evan Chen                  |
| https://github.com/vEnhance/dotfiles/blob/main/py-scripts/tsqx.py |
+-------------------------------------------------------------------+
A = dir 110
C = dir 210
E = dir 330
B = A+C
D = C+E
F = A+E
O 270 = origin
A--C--E--cycle / 0.1 yellow / red
A--B--C--D--E--F--cycle / 0.1 palecyan / blue
O--A / deepgreen
O--C / deepgreen
O--E / deepgreen
*/
\end{asy}
\end{center}

In particular, we have
\[
  \frac{AC \cdot CE \cdot EA}{4 \cdot 3} =
  \opname{Area}(ACE) = \frac{\opname{Area}(ABCDEF)}{2} = \frac 52
\]
and hence
\[ AC \cdot CE \cdot EA = 30. \]
Now $DF = AC$, $FB = CE$, $BD = EA$ so the answer is $30^2 = \boxed{900}$.

\end{document}
