Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

In acute $\triangle A B C$ with centroid $G, A B=22$ and $A C=19$ . Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to $A C$ and $A B$ respectively. Let $G^{\prime}$ be the reflection of $G$ over $B C$ . If $E, F, G$ , and $G^{\prime}$ lie on a circle, compute $B C$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Note that $B, C, E, F$ lie on a circle. Moreover, since $B C$ bisects $G G^{\prime}$ , the center of the circle that goes through $E, F, G, G^{\prime}$ must lie on $B C$ . Therefore, $B, C, E, F, G, G^{\prime}$ lie on a circle. Specifically, the center of this circle is $M$ , the midpoint of $B C$ , as $M E=M F$ because $M$ is the center of the circumcircle of $B C E F$ . So we have $G M=\frac{B C}{2}$ , which gives $A M=\frac{3 B C}{2}$ . Then, by Apollonius's theorem, we have $A B^{2}+A C^{2}=2\left(A M^{2}+B M^{2}\right)$ . Thus $845=5 B C^{2}$ and $B C=13$ .

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