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KMN and PQR are tangent at a fixed point

Source:

March 19, 2013

geometrycircumcirclegeometry unsolvedsimilarity

Problem Statement

Let

ABCD

be cyclic quadrilateral. Let

AC

and

BD

intersect at

R

, and let

AB

and

CD

intersect at

K

. Let

M

and

N

are points on

AB

and

CD

such that

\frac{AM}{MB}=\frac{CN}{ND}

. Let

P

and

Q

be the intersections of

MN

with the diagonals of

ABCD

. Prove that circumcircles of triangles

KMN

and

PQR

are tangent at a fixed point.

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