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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.

Solution

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