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2 pairs of equal angles and points of circumcircle lead to concurrency

Source: Cono Sur Shortlist 2005 G5

July 26, 2019

geometrycircumcircleconcurrencyconcurrent

Problem Statement

Let

O

be the circumcenter of an acute triangle

ABC

and

A_1

a point of the minor arc

AC

of the circle

ABC

. Let

A_2

and

A_3

be points on sides

AB

and

AC

respectively such that

\angle BA_1A_2=\angle OAC

and

\angle CA_1A_3=\angle OAB

. Points

B_2, B_3, C_2

and

C_3

are similarly constructed, with

AC

in

BC, B_3

in

AB, C_2

in

AC

and

C_3

in

BC

. Prove that lines

A_2A_3, B_2B_3

and

C_2C_3

are concurrent.

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