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2 lines concurrent with circumcircle, angle bisectors, circumcircles related

Source: 2014 Oral Moscow Geometry Olympiad grades 10-11 p3

August 9, 2019

geometrycircumcircleangle bisectorconcurrencyconcurrent

Problem Statement

The bisectors

AA_1

and

CC_1

of triangle

ABC

intersect at point

I

. The circumscribed circles of triangles

AIC_1

and

CIA_1

intersect the arcs

AC

and

BC

(not containing points

B

and

A

respectively) of the circumscribed circle of triangle

ABC

at points

C_2

and

A_2

, respectively. Prove that lines

A_1A_2

and

C_1C_2

intersect on the circumscribed circle of triangle

ABC