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 $X$ .

Also notice that $\overline{OA}$ and $\overline{OC}$ are the angle bisectors of angle $\angle BAC$ and $\angle BCA$ respectively. We then deduce $\angle AOC=120^\circ$ .

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

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

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

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 $\pi r^2 = \boxed{\textbf{(B)}\ 12\pi}$ .

~Wilhelm Z

Solution 2

We have $\angle AOC = 120^\circ$ .

Denote by $R$ the circumradius of $\triangle AOC$ . In $\triangle AOC$ , the law of sines implies $2 R = \frac{AC}{\sin \angle AOC} = 4 \sqrt{3} .$

Hence, the area of the circumcircle of $\triangle AOC$ is $\pi R^2 = 12 \pi .$

Therefore, the answer is $\boxed{\textbf{(B) }12 \pi}$ .

~Steven Chen (www.professorchenedu.com)

Solution 3

As in the previous solution, construct the circle that passes through $A$ , $O$ , and $C$ , centered at $X$ . Let $Y$ be the intersection of $\overline{OX}$ and $\overline{AB}$ .

Note that since $\overline{OA}$ is the angle bisector of $\angle BAC$ that $\angle OAC=30^\circ$ . Also by symmetry, $\overline{OX}$ $\perp$ $\overline{AB}$ and $AY = 3$ . Thus $\tan(30^\circ) = \frac{OY}{3}$ so $OY = \sqrt{3}$ .

Let $r$ be the radius of circle $X$ , and note that $AX = OX = r$ . So $\triangle AYX$ is a right triangle with legs of length $3$ and $r - \sqrt{3}$ and hypotenuse $r$ . By Pythagoras, $3^2 + (r - \sqrt{3})^2 = r^2$ . So $r = 2\sqrt{3}$ .

Thus the area is $\pi r^2 = \boxed{\textbf{(B)}\ 12\pi}$ .

-SharpeMind

Solution 5 (SIMPLE)

The semiperimeter is $\frac{6+6+6}{2}=9$ units. The area of the triangle is $9\sqrt{3}$ units squared. By the formula that says that the area of the triangle is its semiperimeter times its inradius, the inradius $r=\sqrt{3}$ . As $\angle{AOC}=120^\circ$ , we can form an altitude from point $O$ to side $AC$ at point $M$ , forming two 30-60-90 triangles. As $CM=MA=3$ , we can solve for $OC=2\sqrt{3}$ . Now, call the center of the circle we are looking for $X$ . From here we can now utilize the fact that since the inscribed angle $\angle{AOC}=120^\circ$ that the arclength that the angle describes is $240^\circ$ . This leaves us $120^\circ$ ( $360-240$ ) left to be formed by the central angle, which by the Central Angle Theorem must be $120^\circ$ as well. Note that now we have a parallelogram with the side $OC$ (which is $2\sqrt{3}$ ) being opposite and therefor equivalent to the radius $XA$ . Now, the area of the circle is just $\pi*(2*\sqrt{3})^2 = 12\pi$ . Select $\boxed{B}$ .

~hastapasta, bob4108, TheHuskyKing

Note For Solution 5

One must be careful as one can easily accidentally get this problem CORRECT by prematurely saying the radius is $2\sqrt{3}$ BEFORE proving that $OC$ is a part of a parallelogram. Otherwise, your statement is simply saying that the Circumcircle's area is $12\pi$ . Meaning that although your answer was correct the reasoning was not.

~TheHuskyKing

Solution 6 (Ptolemy)

Call the diameter of the circle $d$ . If we extend points $A$ and $C$ to meet at a point on the circle and call it $E$ , then $\bigtriangleup OAE=\bigtriangleup OCE$ . Note that both triangles are right, since their hypotenuse is the diameter of the circle. Therefore, $CE=AE=\sqrt{d^2-12}$ . We know this since $OC=OA=OB$ and $OC$ is the hypotenuse of a $30-60-90$ right triangle, with the longer leg being $\frac{6}{2}=3$ so $OC=2\sqrt{3}$ . Applying Ptolemy's Theorem on cyclic quadrilateral $OCEA$ , we get $2({\sqrt{d^2-12}})\cdot{2\sqrt{3}}=6d$ . Squaring and solving we get $d^2=48 \Longrightarrow (2r)^2=48$ so $r^2=12$ . Therefore, the area of the circle is $\boxed{12\pi}$

~Magnetoninja

Video Solution (Just 3 min!)

https://youtu.be/kkm1d09bVpM

~Education, the Study of Everything

Video Solution by TheBeautyofMath

https://www.youtube.com/watch?v=4qgYrCYG-qw&t=1039

~IceMatrix

2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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