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All-Soviet-Union Mathematical Olympiad

Geometry Difficulty 4.6 AIME Prove it Soviet Union

Problem:
Given a circle, diameter $AB$ and a point $C$ on $AB$ , show how to construct two points $X$ and $Y$ on the circle such that (1) $Y$ is the reflection of $X$ in the line $AB$ , (2) $YC$ is perpendicular to $XA$ .

Solution

Solution:
$AB$ is a diameter, so $BX$ is perpendicular to $AX$ and hence parallel to $YC$ . $YX$ is perpendicular to $BC$ , so $YC = YB$ . Hence $X$ and $Y$ lie on the perpendicular bisector of $BC$ .

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