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perpenducilar wanted, circle with diamater an altitude related

Source: 2013 Saudi Arabia GMO TST day I p4

July 31, 2020

geometryperpendicularcirclealtitudes

Problem Statement

In acute triangle

ABC

, points

D

and

E

are the feet of the perpendiculars from

A

to

BC

and

B

to

CA

, respectively. Segment

AD

is a diameter of circle

\omega

. Circle

\omega

intersects sides

AC

and

AB

at

DY

and

G

(other than

A

), respectively. Segment

BE

intersects segments

GD

and

GF

at

X

and

Y

respectively. Ray

DY

intersects side

AB

at

Z

. Prove that lines

XZ

and

BC

are perpendicular

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