MathDB

Midpoints connected to center are perpendicular

Source: China 2016 TST Day 2 Q5

March 16, 2016

geometrycyclic quadrilateral

Problem Statement

Refer to the diagram below. Let

ABCD

be a cyclic quadrilateral with center

O

. Let the internal angle bisectors of

\angle A

and

\angle C

intersect at

I

and let those of

\angle B

and

\angle D

intersect at

J

. Now extend

AB

and

CD

to intersect

IJ

and

P

and

R

respectively and let

IJ

intersect

BC

and

DA

at

Q

and

S

respectively. Let the midpoints of

PR

and

QS

be

M

and

N

respectively. Given that

O

does not lie on the line

IJ

, show that

OM

and

ON

are perpendicular.

Back to Problems View on AoPS