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Source: Federation of Bosnia and Herzegovina, 4th grades, 2008.

May 1, 2008

trigonometrygeometrycircumcirclegeometry proposed

Problem Statement

Given are three pairwise externally tangent circles

K_{1}

,

K_{2}

and

K_{3}

. denote by

P_{1}

tangent point of

K_{2}

and

K_{3}

and by

P_{2}

tangent point of

K_{1}

and

K_{3}

. Let

AB

(

A

and

B

are different from tangency points) be a diameter of circle

K_{3}

. Line

AP_{2}

intersects circle

K_{1}

(for second time) at point

X

and line

BP_{1}

intersects circle

K_{2}

(for second time) at

Y

. If

Z

is intersection point of lines

AP_{1}

and

BP_{2}

prove that points

X

,

Y

and

Z

are collinear.

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