A Triangle and a Circle

A circle overlaps all three sides of a triangle splitting each side into three segments. Find such a triangle, such that the lengths of all nine segments are unique integers.

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.]

Source: Original.

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