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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 9"

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Construct the circle that passes through <math>A</math>, <math>O</math>, and <math>C</math>, centered at <math>X</math>.  
 
Construct the circle that passes through <math>A</math>, <math>O</math>, and <math>C</math>, centered at <math>X</math>.  
  
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Then connect <math>OX</math>, and notice that <math>OX</math> is the perpendicular bisector of <math>AC</math>. Let the intersection of <math>OX</math> with <math>AC</math> be <math>D</math>.
+
Then connect <math>\overline{OX}</math>, and notice that <math>\overline{OX}</math> is the perpendicular bisector of <math>\overline{AC}</math>. Let the intersection of <math>\overline{OX}</math> with <math>\overline{AC}</math> be <math>D</math>.
  
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Also notice that <math>AO</math> and <math>CO</math> are the angle bisectors of angle <math>BAC</math> and <math>BCA</math> respectively. We then deduce <math>AOC=120^\circ</math>.
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Also notice that <math>\overline{OA}</math> and <math>\overline{OC}</math> are the angle bisectors of angle <math>BAC</math> and <math>BCA</math> respectively. We then deduce <math>AOC=120^\circ</math>.
  
 
Consider another point <math>M</math> on Circle <math>X</math> opposite to point <math>O</math>.  
 
Consider another point <math>M</math> on Circle <math>X</math> opposite to point <math>O</math>.  

Revision as of 21:55, 23 November 2021

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 $BAC$ and $BCA$ respectively. We then deduce $AOC=120^\circ$.

Consider another point $M$ on Circle $X$ opposite to point $O$.

As $AOCM$ an inscribed quadrilateral of Circle $X$, $AMC=180^\circ-120^\circ=60^\circ$.

Afterward, deduce that $AXC=2·AMC=120^\circ$.

By the Cosine Rule, we have the equation: (where $r$ is the radius of circle $X$)

$2r^2(1-\cos(120^\circ))=6^2$

$r^2=12$

The area is therefore $12\pi \Rightarrow \boxed{\textbf{(B)}}$.

~Wilhelm Z

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