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Ukrainian National Mathematical Olympiad

Geometry Difficulty 5.1 AIME, harder Prove it Ukraine

Given is circle $\Gamma$ with center in point $O$ and diameter $AB$ . $OBDE$ is a square, $F$ is the second point of intersection of $AD$ and circle $\Gamma$ , $C$ is the middle of the segment $AF$ . Find the value of the angle $OCB$ .

Solution

Since $AB$ is a diameter, then $\angle AFB = 90^\circ$ , and $CO \parallel FB$ as the middle segment. Therefore, $CO \perp CD$ and quadrilateral $OBDC$ is inscribed. Hence, $\angle OCB = \angle ODB = 45^\circ$ .

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