incircle of a right angled triangle

by considering areas, you can establish that the radius of the incircle is
ab/(a + b + c)

by considering equal (bits of) tangents you can also establish that the radius,
 r = 1/2 (b + a - c):

b + a - c  =  (e + g) + (e + f ) - (g + f)  =  2e  =  2r

for integer values of the incircle radius you need a pythagorean triple

with the (subset of) pythagorean triples generated from the shortest side being an odd number

3, 4, 5  has an incircle radius, r = 1
5, 12, 13  has r = 2 (property for shapes where the area value = perimeter value, 'equable')
7, 24, 25  has r = 3
9, 40, 41  has r = 4
etc.

using the (pythagorean triple) property that a chosen odd number ('a') when squared is broken up into two consective integers for 'b' and 'c', you can (using some reasonably complicated algebra for the first relationship above) show that this pattern continues, for ever and ever....

for such pythagorean triples (and the statements are also true in general), you can also establish that
3 must be a factor of either a or b (but not both)
4 is a factor of a or b
5 is a factor of a, b or c