MathDB

Bulgaria 2

Source: BMO Problem 2

May 15, 2005

geometrycircumcircleincenterratiotrigonometrypower of a pointradical axis

Problem Statement

Consider two circles

k_{1},k_{2}

touching externally at point

T

. a line touches

k_{2}

at point

X

and intersects

k_{1}

at points

A

and

B

. Let

S

be the second intersection point of

k_{1}

with the line

XT

. On the arc

\widehat{TS}

not containing

A

and

B

is chosen a point

C

. Let

\ CY

be the tangent line to

k_{2}

with

Y\in k_{2}

, such that the segment

CY

does not intersect the segment

ST

. If

I=XY\cap SC

. Prove that : (a) the points

C,T,Y,I

are concyclic. (b)

I

is the excenter of triangle

ABC

with respect to the side

BC

.

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