Can a triangle and circle be found, such that the radius of the circle is also an integer?
For example, in triangle ABC, the segments would be: AB:(a,b,c), BC:(d,e,f), and CA:(g,h,i), with b, e, and h as chords of the circle. There are probably solutions in which all lengths are positive, as well as solutions in which one of the chords is of length 0 (the circle is tangent to that side of the triangle.) See if you can minimize the perimeter of the triangle. ( For example, a near miss is (6,0,3), (1,8,6), (7,5,4). )
[For a very useful hint, see Theorem 11.16 and its Corollary at http://library.advanced.org/16284/reference_gc_9.htm.]