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angle bisectors and two congruent angles

Source: All-Russian 2021/11.4

April 20, 2021

geometryangle bisectorcircumcircle

Problem Statement

In triangle

ABC

angle bisectors

AA_{1}

and

CC_{1}

intersect at

I

. Line through

B

parallel to

AC

intersects rays

AA_{1}

and

CC_{1}

at points

A_{2}

and

C_{2}

respectively. Let

O_{a}

and

O_{c}

be the circumcenters of triangles

AC_{1}C_{2}

and

CA_{1}A_{2}

respectively. Prove that

\angle{O_{a}BO_{c}} = \angle{AIC}

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