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Median perpendicular to a line

Source: XVII Sharygin Correspondence Round P4

March 2, 2021

geometrycircumcircle

Problem Statement

Let

ABCD

be a square with center

O

, and

P

be a point on the minor arc

CD

of its circumcircle. The tangents from

P

to the incircle of the square meet

CD

at points

M

and

N

. The lines

PM

and

PN

meet segments

BC

and

AD

respectively at points

Q

and

R

. Prove that the median of triangle

OMN

from

O

is perpendicular to the segment

QR

and equals to its half.

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