The three cut off triangles are similar to the original. The inradius is proportional to the size of a triangle. Therefore it suffices to show that the sum of three corresponding sides of the cut off triangles sum to the corresponding side of the original.
Extend the cuts described and they meet at three new points. These points form a new triangle congruent to the original. (There's congruent sides and angles aplenty on my drawing, no need to describe it.)
The two triangles are rotations by 180 about the incenter. The three corresponding sides of the cutoff triangles are now lined up along the corresponding edge.

Posted by Jer
on 20170728 19:28:06 