MathDB

Let

P

be a point inside the acute triangle

ABC,

and let

Q

be the isogonal conjugate of

P.

Let

L,M

and

N

be the midpoints of the shorter arcs

BC,CA

and

AB

of the circumcircle of

ABC,

respectively. Let

X_A

be the intersection of ray

LQ

and circle

(PBC),

let

X_B

be the intersection of ray

MQ

and circle

PCA,

and let

X_C

be the intersection of ray

NQ

and circle

(PAB).

Prove that

P,X_A,X_B

and

X_C

are concyclic or coincide.
Proposed by Gustavo Cruz (São Paulo)

Problem Statement