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Segment equal

Source: All Russian Mathematical Olympiad 2008. 10.3

June 13, 2008

geometrycircumcircleparallelogramanalytic geometrytrigonometrytrapezoidpower of a point

Problem Statement

A circle

\omega

with center

O

is tangent to the rays of an angle

BAC

at

B

and

C

. Point

Q

is taken inside the angle

BAC

. Assume that point

P

on the segment

AQ

is such that

AQ\perp OP

. The line

OP

intersects the circumcircles

\omega_{1}

and

\omega_{2}

of triangles

BPQ

and

CPQ

again at points

M

and

N

. Prove that OM \equal{} ON.

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