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2 perpendicular lines concurrent with circumcircle

Source: Mathematics Regional Olympiad of Mexico Center Zone 2011 P2

November 11, 2021

geometryconcurrencyconcurrentcircumcircle

Problem Statement

Let

ABC

be a triangle and let

L

M

N

be the midpoints of the sides

BC

CA

and

AB

, respectively. The points

P

and

Q

lie on

AB

and

BC

, respectively; the points

R

and

S

are such that

N

is the midpoint of

PR

and

L

is the midpoint of

QS

. Show that if

PS

and

QR

are perpendicular, then their intersection lies on in the circumcircle of triangle

LMN