Can you solve this triangle puzzle

Two triangles have the same perimeter namely 182, the same area namely 546, and the same radius for the inscribed circle namely 6. Both have positive whole numbers for all three side lengths. One is a right triangle, the other is not. Can you find these triangles?

First note that the value of the inradius (6) is unnecessary here,
because it always equals the area (say A) divided by the semiperimeter
(say s). In this case, (546/ (182/2)) = 6, so the inradius can be
calculated from the other given information.

Using Heron's formula we know that

A = sqrt (s (s - a) (s - b) (s - c)). So,

546 = sqrt (91 (91 - a) (91 - b) (91 - c))

Factoring the left side into prime factors, we have

13 * 7 * 3 * 2 = sqrt (91 (91 - a) (91 - b) (91 - c))

Squaring and dividing both sides by 91, we get

13 * 7 * 3 * 3 * 2 * 2 = (91 - a) (91 - b) (91 - c)

Call this equation (A). Note that each of (91 - a), (91 - b) and (91 -
c) is a whole number because a,b,c are given to be whole
numbers. Since 13 is prime, it must divide at least one of these three
numbers. Further, because 91 is divisible by 13, at least one of a,b,c
must be divisible by 13. Suppose (without loss of generality) that a
is divisible by 13. Thus possible values of a are: 13, 26, 39, 52, 65,
78. Of these, we quickly rule out a = 26, because in that case, (91 -
a) = 65 = 13 * 5, so 5 becomes a factor of the right hand side, but it
is not a factor of the left hand side. This leaves the following
possibilities for a: 13, 39, 52, 65, 78. We analyze each of these
separately.

For a = 13, (91 - b) = 91 - (182 - 13 - c) = c - 78, so using (A), we get

7 * 3 * 2 = (c - 78) (91 - c), which gives c = 85 and c =
84. Accordingly, b = 84 and b = 85. Hence, one solution is {13, 84,
85}. This triangle is a right triangle (easily checked using the
Pythagorous theorem).

For a = 39, (91 - b) = 91 - (182 - 39 - c) = c - 52, so using (A), we get