How can I get coordinates and radius of Incirle of the triagle with the length three sides are `a, b, c`?

Question

I used

Insphere[SSSTriangle[5, 7, 9][[1]]]

and got

Sphere[{11/2, Sqrt[11]/2}, Sqrt[11]/2]

But, I cannot get the result when I used

Insphere[SSSTriangle[a, b, c][[1]]]

I only got

Insphere[{{0, 0}, {c, 0}, {(-a^2 + b^2 + c^2)/(2 c), Sqrt[(a + b - c) (a - b + c) (-a + b + c) (a + b + c)]/(2 c)}}]

How can I get the correct result?

Answer 1

FullSimplify providing the information that "the lengths a, b, and c are positive numbers such that each of them is less than the sum of the other two."

assumptions = {a > 0, b > 0, c > 0, a + b > c, a + c > b, b + c > a};

FullSimplify[Insphere[SSSTriangle[a, b, c][[1]]], Assumptions -> assumptions]

Sphere[{1/2 (-a + b + c), 1/2 Sqrt[((a + b - c) (a - b + c) (-a + b + c))/(a + b + c)]},
1/2 Sqrt[((a + b - c) (a - b + c) (-a + b + c))/(a + b + c)]]