MathDB

Observe the beauty

Source: BdMO 2024 Higher Secondary National P9

March 18, 2024

geometryarc midpointInversioncyclic quadrilateralSpiral Similaritypower of a point

Problem Statement

Let

ABC

be a triangle and

M

be the midpoint of side

BC

. The perpendicular bisector of

BC

intersects the circumcircle of

AK

at points

K

and

L

(

K

and

A

lie on the opposite sides of

BC

). A circle passing through

\triangle KMQ

and

M

intersects

AK

at points

P

and

Q

(

P

lies on the line segment

AQ

).

LQ

intersects the circumcircle of

\triangle KMQ

again at

R

. Prove that

BPCR

is a cyclic quadrilateral.

Back to Problems View on AoPS