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rhombus of side \sqrt {DF x EK} wanted (Kyiv City Olympiad 2009 10.4 11.3)

Source:

July 2, 2020

geometryrhombuscircumcircleincenter

Problem Statement

In the triangle

ABC

the angle bisectors

AL

and

BT

are drawn, which intersect at the point

I

, and their extensions intersect the circle circumscribed around the triangle

ABC

at the points

E

and

D

respectively. The segment

DE

intersects the sides

AC

and

BC

at the points

F

and

K

, respectively. Prove that: a) quadrilateral

IKCF

is rhombus; b) the side of this rhombus is

\sqrt {DF \cdot EK}

.
(Rozhkova Maria)