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show that PQ is diameter

Source: Middle European Mathematical Olympiad 2013 T-6

May 17, 2014

geometrycircumcircletrigonometrygeometric transformationreflectionperpendicular bisectorpower of a point

Problem Statement

Let

K

be a point inside an acute triangle

ABC

, such that

BC

is a common tangent of the circumcircles of

AKB

and

F

. Let

D

be the intersection of the lines

CK

and

AB

, and let

E

be the intersection of the lines

BK

and

AC

. Let

F

be the intersection of the line

BC

and the perpendicular bisector of the segment

DE

. The circumcircle of

ABC

and the circle

k

with centre

F

and radius

FD

intersect at points

P

and

Q

. Prove that the segment

PQ

is a diameter of

k

.

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