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midpoint, related to perpendiculars, circumcircle - Austrian-Polish 1999

Source:

May 19, 2019

geometrycircumcirclemidpointperpendicular bisectorperpendicular

Problem Statement

Let

S

be points on the same side of a line

g

in the plane. Let

M

and

N

be the feet of the perpendiculars from

P

and

Q

to

g

respectively. Point

S

lies between the lines

PM

and

QN

and satisfies and satisfies

PM = PS

and

QN = QS

. The perpendicular bisectors of

SM

and

SN

meet in a point

R

. If the line

RS

intersects the circumcircle of triangle

PQR

again at

T

, prove that

S

is the midpoint of

RT

.

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