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Three circles with some intersections

Source: Germany 2013 - Problem 3

December 5, 2022

geometrycirclestangent

Problem Statement

Given two circles

k_1

and

k_2

which intersect at

Q

and

Q'.

Let

P

be a point on

k_2

and inside of

k_1

such that the line

PQ

intersects

k_1

in a point

X\ne Q

and such that the tangent to

k_1

at

X

intersects

k_2

in points

A

and

B.

Let

k

be the circle through

A,B

which is tangent to the line through

P

parallel to

AB.

Prove that the circles

k_1

and

k

are tangent.

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