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 2 of 6: Recall the Miquel point of ABCD lies on KL

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E=\text{Miquel point of }ABCD\implies E\in KL

Detailed analysis

Let $E$ be the Miquel point of the complete quadrilateral formed by lines $AB,BC,CD,DA$ , that is, the common second intersection point of circles $(KBC)$ , $(KAD)$ , $(LAB)$ , $(LCD)$ . It is a standard fact that this Miquel point $E$ lies on the line $KL$ joining the two diagonal points.