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Nice Miquel-like geometry

Source: Russian TST 2020, Day 6 P3

March 21, 2023

geometry

Problem Statement

In a convex quadrilateral

ABCD

, the lines

AB

and

DC

intersect at point

P{}

and the lines

AD

and

BC

intersect at point

Q{}

. The points

E{}

and

F{}

are inside the quadrilateral

ABCD

such that the circles

(ABE), (CDE), (BCF),(ADF)

intersect at one point

K{}

. Prove that the circles

(PKF)

and

(QKE)

intersect a second time on the line

PQ

.

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