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center of circle tangent internally to 2 intersecting circles, lies on bisector

Source: 2021 Ukraine NMO 9.6 10.6

April 4, 2021

geometrytangent circles

Problem Statement

Circles

w_1

and

w_2

intersect at points

P

and

Q

and touch a circle

w

with center at point

O

internally at points

A

and

B

, respectively. It is known that the points

A,B

and

Q

lie on one line. Prove that the point

O

lies on the external bisector

\angle APB

.
(Nazar Serdyuk)