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Tangents to circumcircles in a triangle

Source: Germany 2012 - Problem 3

December 6, 2022

geometrycircumcirclecircletangentTriangle

Problem Statement

Let

ABC

a triangle and

k

a circle such that: (1) The circle

k

passes through

A

and

B

and touches the line

AC.

(2) The tangent to

k

at

B

intersects the line

AC

in a point

X\ne C.

(3) The circumcircle

\omega

of

BXC

intersects

k

in a point

Q\ne B.

(4) The tangent to

\omega

at

X

intersects the line

AB

in a point

Y.

Prove that the line

XY

is tangent to the circumcircle of

BQY.

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