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4 tangents to 2 ext. tangent circles (2 each) are also tangent to a third circle

Source: 2014 Sharygin Geometry Olympiad Final Round 8.6

August 3, 2018

geometrytangent circlesTangents

Problem Statement

Two circles

k_1

and

k_2

with centers

O_1

and

O_2

are tangent to each other externally at point

O

. Points

X

and

Y

k_1

and

k_2

respectively are such that rays

O_1X

and

O_2Y

are parallel and codirectional. Prove that two tangents from

X

k_2

and two tangents from

Y

k_1

touch the same circle passing through

O

.
(V. Yasinsky)