MathDB

In a triangle

ABC

the bisectors through vertices

B

and

C

meet the sides

\left [ AC \right ]

and

\left [ AB \right ]

D

and

E

respectively. Let

I_{c}

be the center of the excircle which is tangent to the side

\left [ AB \right ]

and

F

the midpoint of

\left [ BI_{c} \right ]

. If

\left | CF \right |^2=\left | CE \right |^2+\left | DF \right |^2

, show that

ABC

is an equilateral triangle.

Problem Statement