MathDB

Geometry Problem

Source: Azerbaijan Math Olympiad Training

December 15, 2019

geometryTST

Problem Statement

Consider two circles

k_1,k_2

touching at point

T

. A line touches

k_2

at point

X

and intersects

k_1

at points

A,B

where

B

lies between

A

and

X

.Let

S

be the second intersection point of

k_1

with

XT

. On the arc \overarc{TS} not containing

A

and

B

, a point

C

is choosen. Let

CY

be the tangent line to

k_2

with

Y\in{k_2}

, such that the segment

CY

doesn't intersect the segment

ST

.If

I=XY\cap{SC}

, prove that :

(a)

the points

C,T,Y,I

are concyclic.

(b)

I

is the

A-excenter

of

\triangle ABC

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