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Prove that the incenters of triangles $PEF, PCA$, and $PBA$ are collinear.

Source: Moldova TST 2019

January 30, 2023

circumcirclegeometryincenter

Problem Statement

The circle

\Omega

with center

O

is circumscribed to acute triangle

ABC

. Let

P

be a point on the circumscribed circle of

OBC

, such that

P

is inside

ABC

and is different from

B

and

C

. Bisectors of angles

BPA

and

CPA

intersect the sides

AB

and

AC

in points

E

and

F.

Prove that the incenters of triangles

PEF, PCA

and

PBA

are collinear.