Question 666171
Let a,b,c be sides of a triangle such that:

a=b or b=c or a=c but not a=b=c (that would be equilateral)

In addition we have that a+b>c  or a+c > b  or b+c > a (by triangle inequality)

Notice that if two sides are equal, then if the other side is less, it's always a triangle. The only way we don't have a triangle is if we have a,b be our equal sides and c be the longest side and  c>2a. So 2a < c.

Let a=b.

c must be less than 15.

So a must be 8 or 9.

a can't be 10 because then it is equilateral.

a can be 11 through 14 since c will be 2 through 8.

So a =  8,9,11,12,13,14.

So we have 6 isosceles triangles with perimeter 30.

8,8,14
9,9,12
11,11,8
12,12,6
13,13,4
14,14,2