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Another orthocenter problem <3

Source: Mexico National Olympiad 2019 P6

November 12, 2019

geometrycircumcircle

Problem Statement

Let

ABC

be a triangle such that

\angle BAC = 45^{\circ}

. Let

H,O

be the orthocenter and circumcenter of

ABC

, respectively. Let

\omega

be the circumcircle of

ABC

and

P

the point on

\omega

such that the circumcircle of

PBH

is tangent to

BC

. Let

X

and

Y

be the circumcenters of

PHB

and

PHC

respectively. Let

O_1,O_2

be the circumcenters of

PXO

and

PYO

respectively. Prove that

O_1

and

O_2

lie on

AB

and

AC

, respectively.

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