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Parallel induced by intersections with circles

Source: Mexico National Olympiad 2017, Problem 3

November 6, 2017

geometryorthocenterparallel

Problem Statement

Let

ABC

be an acute triangle with orthocenter

H

. The circle through

HB

, and

C

intersects lines

AB

and

AC

at

D

and

E

respectively, and segment

DE

intersects

HB

and

HC

at

P

and

Q

respectively. Two points

X

and

Y

, both different from

A

, are located on lines

AP

and

AQ

respectively such that

X, H, A, B

are concyclic and

Y, H, A, C

are concyclic. Show that lines

XY

and

BC

are parallel.

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