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Geo with complete quadrilateral

Source: Russian TST 2016, Day 11 P1 (Group NG)

April 19, 2023

geometrycyclic quadrilateral

Problem Statement

In the cyclic quadrilateral

ABCD

, the diagonal

BD

is divided in half by the diagonal

AC

. The points

E, F, G

and

H{}

are the midpoints of the sides

AB, BC, CD{}

and

DA

respectively. Let

P = AD \cap BC

and

Q = AB \cap CD{}

. The bisectors of the angles

APC

and

AQC

intersect the segments

EG

and

FH

at the points

X{}

and

Y{}

respectively. Prove that

XY \parallel BD

.

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