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3 tangent circles, angles chasing

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1995 p5

February 20, 2020

anglestangent circlesgeometry

Problem Statement

Let

A,B

be points on a circle

k

with center

S

such that

\angle ASB = 90^o

. Circles

k_1

and

k_2

are tangent to each other at

Z

and touch

k

A

and

B

respectively. Circle

k_3

inside

\angle ASB

is internally tangent to

k

C

and externally tangent to

k_1

and

k_2

X

and

Y

, respectively. Prove that

\angle XCY = 45^o