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, 2022

Geometry Difficulty 5.8 AIME, harder Prove it United States

Problem:

Let triangle $A B C$ be an acute triangle with circumcircle $\Gamma$ . Let $X$ and $Y$ be the midpoints of minor arcs $\widehat{A B}$ and $\widehat{A C}$ of $\Gamma$ , respectively. If line $X Y$ is tangent to the incircle of triangle $A B C$ and the radius of $\Gamma$ is $R$ , find, with proof, the value of $X Y$ in terms of $R$ .

Solution

Solution:

Note that $X$ and $Y$ are the centers of circles $(A I B)$ and $(A I C)$ , respectively, so we have $X Y$ perpendicularly bisects $A I$ , where $I$ is the incenter. Since $X Y$ is tangent to the incircle, we have $A I$ has length twice the inradius. Thus, we get $\angle A = 60^{\circ}$ . Thus, since $\widehat{X Y} = \frac{\widehat{B A C}}{2}$ , we have $\widehat{X Y}$ is a $120^{\circ}$ arc. Thus, we have $X Y = R \sqrt{3}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.