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Three circles tangent to everything

Source: Turkey TST 2022 P4 Day 2

March 13, 2022

geometrygeometry proposedcirclestangent circles

Problem Statement

We have three circles

w_1

,

w_2

and

\Gamma

at the same side of line

l

such that

w_1

and

w_2

are tangent to

l

at

K

and

L

and to

\Gamma

at

M

and

N

, respectively. We know that

w_1

and

w_2

do not intersect and they are not in the same size. A circle passing through

K

and

L

intersect

\Gamma

at

A

and

B

. Let

R

and

S

be the reflections of

M

and

N

with respect to

l

. Prove that

A, B, R, S

are concyclic.

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