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Cyclic Quadrilateral, bisectors, points lie on perpendicular line

Source: 54 Polish MO 2003 Second Round - First Day Problem 2

February 19, 2018

geometrycyclic quadrilateralPolandbisectorperpendiculargeometry unsolved

Problem Statement

The quadrilateral

ABCD

is inscribed in the circle

o

. Bisectors of angles

DAB

and

ABC

intersect at point

P

, and bisectors of angles

BCD

and

CDA

intersect in point

Q

. Point

M

is the center of this arc

BC

of the circle

o

which does not contain points

D

and

A

. Point

N

is the center of the arc

DA

of the circle

o

, which does not contain points

B

and

C

. Prove that the points

P

and

Q

lie on the line perpendicular to

MN

.

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