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concurrency wanted, line secant to both intersecting circles related

Source: 2015 Croatia MO p3

August 3, 2020

concurrencyconcurrentgeometrycircles

Problem Statement

Circles

k_1

and

k_2

intersect at points

M

and

N

. The line

\ell

intersects the circle

k_1

at points

A

and

C

, the circle

K_2

at points

B

and

D

so that the points

A,B,C

and

D

lie on the line

\ell

are in that order. Let

X

a point on the line

MN

such that the point

M

is located between the points

X

and

N

. Let

P

be the intersection of lines

AX

and

BM

, and

Q

be the intersection of lines

DX

and

CM

. If

K

is the midpoint of segment

AD

and

L

is the midpoint of segment

BC

, prove that the lines

XK

and

ML

intersect on the line

PQ

.

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