2021 Fall AMC 12B Problems/Problem 9

Problem

Triangle $ABC$ is equilateral with side length $6$ . Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$ , $O$ , and $C$ ?

$\textbf{(A)} \: 9\pi \qquad\textbf{(B)} \: 12\pi \qquad\textbf{(C)} \: 18\pi \qquad\textbf{(D)} \: 24\pi \qquad\textbf{(E)} \: 27\pi$

Solution 1 (Cosine Rule)

Construct the circle that passes through $A$ , $O$ , and $C$ , centered at $X$ .

Then connect $\overline{OX}$ , and notice that $\overline{OX}$ is the perpendicular bisector of $\overline{AC}$ . Let the intersection of $\overline{OX}$ with $\overline{AC}$ be $D$ .

Also notice that $\overline{OA}$ and $\overline{OC}$ are the angle bisectors of angle $\angleBAC$ (Error compiling LaTeX. Unknown error_msg) and $\angleBCA$ (Error compiling LaTeX. Unknown error_msg) respectively. We then deduce $AOC=120^\circ$ .