Track / Stage 5 / 94 of 400 #694 of 1964

Problem 694

AIME late

Geometry Difficulty 5.2 Prove it

[ Intersecting Circles $]$ [Symmetry helps solve the problem.]

Two equal circles with centers $O_{1}$ and $O_{2}$ intersect at points $A$ and $B$ . The segment $O_{1} O_{2}$ intersects these circles at points $M$ and $N$ .

Prove that the quadrilaterals $O_{1} A O_{2} B$ and $A M B N$ are rhombuses.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

The sides of the quadrilateral $O_{1} A O_{2} B$ are equal to the radii of the circles. The diagonals of the quadrilateral $A M B N$ are perpendicular to each other and are bisected by the point of intersection.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.