How many different non-congruent isosceles triangles can be formed by connecting three of the dots in a 4x4 square array of dots like the one shown below?
[asy]
size(50);
dot((0,0));dot((0,1));dot((0,2));dot((0,3));
dot((1,0));dot((1,1));dot((1,2));dot((1,3));
dot((2,0));dot((2,1));dot((2,2));dot((2,3));
dot((3,0));dot((3,1));dot((3,2));dot((3,3));
[/asy]
Two triangles are congruent if they have the same traced outline, possibly up to rotating and flipping. This is equivalent to having the same set of 3 side lengths.
We have the following list of triangles:
(2,2,sqrt(10))
(sqrt(10),sqrt(10),sqrt(2))
(2*sqrt(2),2*sqrt(2),3*sqrt(2))
(sqrt(2),sqrt(2),2)
(1,1,sqrt(2))
(2*sqrt(2),2*sqrt(2),2)
(2*sqrt(2),2*sqrt(2),2*sqrt(2))
(sqrt(5),sqrt(5),sqrt(10))
(sqrt(2),sqrt(2),sqrt(2))
(sqrt(5),sqrt(5),sqrt(2))
(sqrt(5),sqrt(5),2)
(sqrt(10),sqrt(10),sqrt(10))
(sqrt(5),sqrt(5),sqrt(5))
So there are 13 possible triangles.
