MathDB

Lots of Cyclic Quads

Source: 2018 USAMO #5

April 19, 2018

geometrycyclic quadrilateralUSAMOMiquel pointprism lemmaUSA(J)MOHi

Problem Statement

In convex cyclic quadrilateral

ABCD

, we know that lines

AC

and

BD

intersect at

E

, lines

AB

and

CD

intersect at

F

, and lines

BC

and

DA

intersect at

G

. Suppose that the circumcircle of

\triangle ABE

intersects line

CB

at

B

and

P

, and the circumcircle of

\triangle ADE

intersects line

CD

at

D

and

Q

, where

C,B,P,G

and

C,Q,D,F

are collinear in that order. Prove that if lines

FP

and

GQ

intersect at

M

, then

\angle MAC = 90^\circ

.
Proposed by Kada Williams

Back to Problems View on AoPS