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incircle, circumcircles and 3 orthogonal circles

Source: Czech and Slovak Match 2000 P2

October 1, 2017

geometryorthogonal circlescircumcircle

Problem Statement

Let

{ABC}

be a triangle,

{k}

its incircle and

{k_a,k_b,k_c}

three circles orthogonal to

{k}

passing through

{B}

and

{C, A}

and

{C}

, and

{A}

and

{B}

respectively. The circles

{A'}

meet again in

{C'}

; in the same way we obtain the points

{B'}

and

{A'}

. Prove that the radius of the circumcircle of

{A'B'C'}

is half the radius of

{k}

.

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