There are two cases to consider:
Case 1: C is on the horizontal line through A or the vertical line through B.
Without loss of generality, suppose C is on the horizontal line through A. If AB is a horizontal segment, then △ABC cannot be isosceles.
Otherwise, AC=AB if and only if C is one of the two points on this horizontal line that are the same distance from A as B is.
The same argument applies if C is on the vertical line through B. Thus, there are at most 2+2=4 such points C.
Case 2: C is not on the horizontal line through A or the vertical line through B.
Then △ABC is isosceles if and only if C lies on the perpendicular bisector of AB. There are 4 such points.
Hence, there are 4+4=8 such points C.
Combining like terms, we have the inequality 2x2+nx−135<0. We can use the quadratic formula to solve for the roots of 2x2+nx−135=0.
The quadratic formula gives us:
Since we are given that the inequality has no real solutions, the discriminant n2+1080 must be negative. This gives us the inequality n2<−1080.
Since n is an integer, the only possible values of n are −34, −33, −32, ..., 32, 33, and 34. Therefore, there are 67 possible values of n.