Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Prove it Soviet Union

Problem:

$E$ is a point on the diagonal $BD$ of the square $ABCD$ . Show that the points $A$ , $E$ and the circumcenters of $ABE$ and $ADE$ form a square.

Solution

Solution:

Let $O$ , $O'$ be the circumcenters of $ABE$ , $ADE$ respectively. Then $OA = OE$ and $\angle AOB = 2 \angle ABE = 90^\circ$ . Similarly, $O'A = O'E$ and $\angle AOE = 2\angle ADE = 90^\circ$ . Hence $AOEO'$ is a square.

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