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equal products of distances,cyclic ABCD, tangents concurrent with symmetric line

Source: 2006 Oral Moscow Geometry Olympiad grades 10-11 p4

October 20, 2020

geometrySymmetricdistancecyclic quadrilateral

Problem Statement

The quadrangle

ABCD

is inscribed in a circle, the center

O

of which lies inside it. The tangents to the circle at points

A

and

C

and a straight line, symmetric to

BD

wrt point

O

, intersect at one point. Prove that the products of the distances from

O

to opposite sides of the quadrilateral are equal.
(A. Zaslavsky)