Finding coordinates of two intersecting circle

Question

I have coordinates of two points P1 and P2 and the radius of circle R and what I need, is to find coordinates of two posible circle centers S1 and S2.

example intersecting circle

When I'm trying to use

(X-Xo)^2+(Y-Yo)^2=R^2

I'm getting something like this

Y^2(1+(P1x-P2x)^2/(P1y-P2y)^2)+ y((P1x-P2x)^2/4+(P1x-P2x)^2/(P1y-P2y)^2)+R^2+1/(P1x-P2x)^2+(P1x-P2x)^2/4+(P1y-P2y)^2/4

When I'm trying to use vectors I'm getting something similarly long and none of my equations gives proper values.

I know that there have to be some smart solution but I can't find it. Please help

Answer 1

The centers will lie on the perpendicular bisector of the line segment joining the intersection points. Use the Pythagorean theorem to find the distance of those centers from the midpoint of that line segment.

That said, the system of equations that you’ve set up isn’t all that long or difficult. Try solving it directly—it builds character.

Answer 2

Helo You can find answer by finding vector that points center of circle from the point between the P1 and P2. This vector will be perpendicular to the vector that links points.

If you make following variable substitution, you get short expresion for the vector.

a=P2x-P1x

b=P2y-P1y

solution

S1x=P1x+a/2+x(of vector)

S1y=P1y+b/2+y(of vector)

There are two solutions so second is for S2.