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Config geo with circle of diameter the median

Source: Cono Sur 2023 P5

August 8, 2023

geometry

Problem Statement

Let

ABC

be an acute triangle and

D, E, F

are the midpoints of

BC, CA, AB

, respectively. The circle with diameter

BC

intersects the lines

AB

and

AC

at points

P

and

Q

, respectively. The lines through

P

and

Q

parallel to

BC

intersect

DE

at point

R

and

DF

at point

S

, respectively. The circumcircle of

DPR

intersects

AB

at

X

, the circumcircle of

DQS

intersects

AC

in

Y

, and these two circles intersect again point

Z

. Prove that

Z

is the midpoint of

XY

.

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