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Tangents parallel to tangents

Source: Sharygin Geometry Olympiad Correspondence Round 2017 P-13 (Grade-9-11)

May 1, 2017

geometry

Problem Statement

Two circles pass through points

A

and

B

. A third circle touches both these circles and meets

AB

at points

C

and

D

. Prove that the tangents to this circle at these points are parallel to the common tangents of two given circles. Proposed by A.Zaslavsky