Answer:
diameter = m - c
Explanation:
In ΔABC, let ∠C be the right angle. The length of the tangents from point C to the inscribed circle are "r", the radius. Then the lengths of tangents from point A are (b-r), and those from point B have length (a-r).
The sum of the lengths of the tangents from points A and B on side "c" is ...
(b-r) +(a-r) = c
(a+b) -2r = c
Now, the problem statement defines the sum of side lengths as ...
a+b = m
and, of course, the diameter (d) is 2r, so we can rewrite the above equation as ...
m -d = c
m - c = d . . . . add d-c
The diameter of the inscribed circle is the difference between the sum of leg lengths and the hypotenuse.