Problem 5

Line

\ell

intersects sides

BC

and

AD

of cyclic quadrilateral

ABCD

at its interior points

R

and

S

, respectively, and intersects ray

DC

beyond point

C

Q

, and ray

BA

beyond point

A

P

. The circumcircles of triangles

QCR

and

QDS

intersect at

N\neq Q

, while the circumcircles of triangles

PAS

and

PBR

intersect at

M\neq P

. Let lines

MP

and

NQ

meet at point

X

, lines

AB

and

CD

meet at point

K

, and lines

BC

and

AD

meet at point

L

. Prove that point

X

lies on line

KL

Step 1 of 6: M, N, P, Q lie on one circle

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M,N,P,Q\text{ are concyclic}

Detailed analysis

Point $M$ is the Miquel point of lines $AB$ , $\ell$ , $AD$ , $BC$ restricted to the configuration around $P,S,A,R,B$ , and $N$ is the Miquel point of lines $CD$ , $BC$ , $\ell$ , $AD$ around $Q,R,C,S,D$ . Using directed angles and the cyclic quadrilaterals $QNRC$ , $PMAS$ , $ABCD$ , one computes $\angle NMP=\angle NRQ+\angle DAB=\angle NRQ+\angle QNR=\angle NQP$ , so $M,N,Q,P$ are concyclic.