Upon further review of the problem (so about ten minutes after I posted and had walked out the door), I realized that my equation is really just a special case of Hero's Formula. This is because the semi perimeter of the triangle minus one of the sides is equal to the radius of the circle on the opposite side of the side. So
$ S=\frac{2(a+b+c)}{2}=a+b+c $
Hero's Formula:
$ A=\sqrt{(S)(S-(a+b))(S-(a+c))(S-(b+c))} $
Now do simple subtraction to find $ S-(a+b) $
$ S-(a+b)=a+b+c-a-b=c $
In a similar manner we find $ S-(a+c)=b $ and $ S-(b+c)=c $
Substituting:
$ A=\sqrt{(S)(a)(b)(c)}=\sqrt{abc(a+b+c)} $ Jhunsber
