Problem #PRU-5192

Problem

Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle $EFG$ the line $EH$ is the median. Two circles with centres $A$ and $C$ are inscribed into triangles $EFH$ and $EGH$ respectively, they are tangent to the median $EH$ at the points $B$ and $D$. Find the length of $BD$ if $EF-EG=2$.