Concurrence of the Median, a Chord and a Diameter of the Incircle

In the triangle

ABC

, let the incircle have center

I

and let its points of tangency to the sides

BC

,

CA

and

AB

be points

D

,

E

and

F

, respectively. Let

M

be the midpoint of

BC

. Then the lines

AM

,

EF

and

DI

are concurrent. Drag the locators

A

,

B

and

C

to see that this holds for any triangle.