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Arbitary point on nine-point circle and cyclic quadrilaterals

Source: 2021 Macedonian Balkan MO TST - Problem 1

August 18, 2021

geometrycyclic quadrilateral

Problem Statement

Let

ABC

be an acute triangle. Let

D

,

E

and

F

be the feet of the altitudes from

A

,

B

and

C

respectively and let

H

be the orthocenter of

\triangle ABC

. Let

X

be an arbitrary point on the circumcircle of

\triangle DEF

and let the circumcircles of

\triangle EHX

and

\triangle FHX

intersect the second time the lines

CF

and

BE

second at

Y

and

Z

, respectively. Prove that the line

YZ

passes through the midpoint of

BC

.

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