2006 Gauss 8 Problem 23

In the diagram, the points are evenly spaced vertically and horizontally. A segment AB is drawn using two of the points, as shown. Point C is chosen to be one of the remaining 18 points. For how many of these 18 possible points is triangle ABC isosceles?

\textbf{(A)}\ 6\quad \textbf{(B)}\ 4\quad \textbf{(C)}\ 5\quad \textbf{(D)}\ 2\quad \textbf{(E)}\ 3

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.