Track / Stage 4 / 286 of 340 #546 of 1964

Problem 546

AMC 12 late, AIME early

Geometry Difficulty 4.9 Prove it

Two circles intersect at points $A$ and $B$ ; $M N$ is a common tangent to them. Prove that the line $A B$ bisects the segment $M N$ .

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

3.10. Let $O$ be the point of intersection of the line $A B$ and the segment $M N$ . Then $O M^{2}=$ $=O A \cdot O B=O N^{2}$ , i.e., $O M=O N$ .

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