Let $I$ be the incenter of a triangle $ABC$ and $M$ be the mid-point of the side $BC$.
If the line $IM$ cuts the height $AH$ in the point $E$, show that $AE=r$, where $r$ is the radius of the circle inscribed in $ABC$.

I tried to solve this using "classical" geometry and using analytical geometry and failed.
I even drew it in Geogebra as follows:
[![Drawing](https://i.stack.imgur.com/x0Syc.jpg)](https://i.stack.imgur.com/x0Syc.jpg)

Can someone help?

Thanks