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More intersections and cyclic quadrilaterals

Source: 2021 Centroamerican and Caribbean Mathematical Olympiad, P6

August 12, 2021

cyclic quadrilateralgeometry

Problem Statement

Let

ABC

be a triangle with

AB<AC

and let

M

be the midpoint of

AC

. A point

P

(other than

B

) is chosen on the segment

BC

in such a way that

AB=AP

. Let

D

be the intersection of

AC

with the circumcircle of

\bigtriangleup ABP

distinct from

A

, and

E

be the intersection of

PM

with the circumcircle of

\bigtriangleup ABP

distinct from

P

. Let

K

be the intersection of lines

AP

and

DE

. Let

F

be a point on

BC

(other than

P

) such that

KP=KF

. Show that

C,\ D,\ E

and

F

lie on the same circle.

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