Because the area of a triangle is directly proportional to the length and height of the base, to minimize the area of a triangle, the length of the base must be minimized, even if A is the smallest.

The minimum value is (a+c)/2, because three sides of a triangle form a arithmetic progression.

According to Pythagorean theorem, A 2+B 2 = C 2 can be obtained, so A = (C 2-B 2)/2c.

Because of b>0, the minimum value is [(c 2-b 2)/2c+c]/2 = c/2.

Therefore, the minimum triangle area is c/2.