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Quadrilateral geo

Source: IGO 2022 Intermediate P5

December 13, 2022

geometry

Problem Statement

Let

ABCD

be a quadrilateral inscribed in a circle

\omega

with center

O

. Let

P

be the intersection of two diagonals

AC

and

BD

. Let

Q

be a point lying on the segment

OP

. Let

E

and

F

be the orthogonal projections of

Q

on the lines

AD

and

BC

, respectively. The points

M

and

NF

lie on the circumcircle of triangle

QEF

such that

QM \parallel AC

and

QN \parallel BD

. Prove that the two lines

ME

and

NF

meet on the perpendicular bisector of segment

CD

.
Proposed by Tran Quang Hung, Vietnam

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