3t,5t-12, and t+20

The triangle will be isosceles if any two of these are equal, and
the resulting sides form a triangle.

Try 

 3t = 5t-12
-2t = -12
  t = 6

Then the sides would be

3t = 3(6) = 18
5t-12 = 5(6)-12 = 18
t + 20 = 26

That's a possible isosceles triangle because the sum of any 
two sides is greater than the third side, and two of the sides
are equal in length.


Try 

  3t = t + 20 
  2t = 20
   t = 10

Then the sides would be

3t = 3(10) = 30
5t-12 = 5(10)-12 = 50-12 = 38  
t+20 = 10+20 = 30    

That's a possible isosceles triangle because the sum of any 
two sides is greater than the third side, and two of the sides
are equal in length.


Try 

  5t - 12 = t + 20 
       4t = 32
        t = 8

Then the sides would be

3t = 3(8) = 24
5t-12 = 5(8)-12 = 40-12 = 28  
t+20 = 8+20 = 28    

That's a possible isosceles triangle because the sum of any 
two sides is greater than the third side, and two of the sides
are equal in length.
 
----

These are the only possible isosceles triangles.  Since an equilateral
triangle is isosceles and none of these possibilities were equilateral
there are no equilateral triangles possible.

Edwin