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equidistant points from midpoint,orthocenter (2022 Novosibirsk Oral Geo Oly 9.7)

Source:

September 3, 2023

geometryorthocenterequal segments

Problem Statement

Altitudes

AA_1

and

CC_1

of an acute-angled triangle

ABC

intersect at point

H

. A straight line passing through point

H

parallel to line

A_1C_1

intersects the circumscribed circles of triangles

AHC_1

and

CHA_1

at points

X

and

Y

, respectively. Prove that points

X

and

Y

are equidistant from the midpoint of segment

BH