But a more geometric method and, I think, simpler is this: Because the two circles have the same radius, the line between the two points of intersection is the perpendicular bisector of the line between the two centers. Knowing the coordinates of the centers, you can find the coordinates of the midpoint and the slope, m, of the line between centers. The perpendicular bisector is the line through that midpoint with slope -1. Find the points at which that line intersects either circle.
In fact, here is what I woud do: Given that one circle has center (x0,y0) and radius r and the other has center (x1, y1) and radius r, I would first translate so that the first circle has center at (0,0), by subtracting x0 from every x coordinate and y0 from every y coordinate. That means that the second circle has center at (a, b) where a= x1- x0 and b= y1- y0. The line between centers is now just y= (b/a)x, which has slope b/a, and the midpoint is (a/2, b/2). The perpendicular bisector is given by y= (-a/b)x. Also the first circle now has equation so the line y= (-b/a)x will intersect the first circle where . Now and . Of course, y, then, is , using the same sign as the corresponding y: and .
Finally, translate back by adding x0 to the x coordinate and y0 to the y coordinate.