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Lots of concurrent lines!

Source: Brazil EGMO TST2 2023 #3

January 29, 2024

geometrycircumcircleCircumcenterincenterincircleconcurrency

Problem Statement

Let

\Delta ABC

be a triangle and

L

be the foot of the bisector of

\angle A

. Let

O_1

and

O_2

be the circumcenters of

\triangle ABL

and

\triangle ACL

respectively and let

B_1

and

C_1

be the projections of

C

and

B

through the bisectors of the angles

\angle B

and

\angle C

respectively. The incircle of

\Delta ABC

touches

AC

and

AB

at points

B_0

and

C_0

respectively and the bisectors of angles

\angle B

and

\angle C

meet the perpendicular bisector of

AL

at points

Q

and

P

respectively. Prove that the five lines

PC_0, QB_0, O_1C_1, O_2B_1

and

BC

are all concurrent.

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