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<OHC = 90^o wanted, XP + Y Q = AB + XY ,orthocenter, circumcenter

Source: 2019 Grand Duchy of Lithuania, Mathematical Contest p3 (Baltic Way TST)

October 3, 2020

geometryright angleperpendicularorthocentercircumcircle

Problem Statement

Let

ABC

be an acute triangle with orthocenter

H

and circumcenter

O

. The perpendicular bisector of segment

CH

intersects the sides

AC

and

BC

in points

X

and

Y

, respectively. The lines

XO

and

YO

intersect the side

AB

in points

P

and

Q

, respectively. Prove that if

XP + Y Q = AB + XY

then

\angle OHC = 90^o