Given the radius and x,y coordinates of the center point of two circles how can I calculate their points of intersection if they have any?

**Answer**

This can be done without any trigonometry at all. Let the equations of the circles be

(x−x1)2+(y−y1)2=r21,

(x−x2)2+(y−y2)2=r22.

By subtracting (2) from (1) and then expanding, we in fact obtain a *linear* equation for x and y; after a little rearranging it becomes

−2x(x1−x2)−2y(y1−y2)=(r21−r22)−(x21−x22)−(y21−y22).

(If the circles intersect, this is the equation of the line that passes through the intersection points.) This equation can be solved for one of x or y; let’s suppose y1−y2≠0 so that we can solve for y:

y=−x1−x2y1−y2x+….

Substituting this expression for y into (1) or (2) gives a quadratic equation in only x. Then the x-coordinates of the intersection points are the solutions to this; the y-coordinates can be obtained by plugging the x-coordinates into (3).

**Attribution***Source : Link , Question Author : Joe Elder , Answer Author : Don Hatch*