Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.4 AIME Prove it United States

Problem:
Let $\overline{S T}$ be a chord of a circle $\omega$ which is not a diameter, and let $A$ be a fixed point on $\overline{S T}$ . For which point $X$ on minor arc $\widehat{S T}$ is the length $A X$ minimized?

Solution

Solution:
Extend the circular segment to make a whole circle, and let $O$ be its center. Draw $O A$ and let it meet the circle at $X$ . Then the circle with center $A$ and radius $A X$ is tangent to the larger circle at $A$ , and thus lies entirely inside it. Therefore, the distance from $A$ to any other point on the circular arc is greater than $A X$ , so $X$ is the desired point.

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