How do I get get x and y position of a particular location on two intersecting circles (Vesica Pisces)?

I have the radius and center $(x,y)$ on both circles, but how do I get the $(x,y)$ of the red circle, or in other words how do I get the $(x,y)$ position of where the circles intersect at the top or bottom?

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You can form a right triangle with the hypotenuse as the line between the center of one circle and the red dot (the length is the radius of the circles). The base of the triangle is half the distance between the centers of the two circles. You can then solve for the third side. – axblount Aug 31 '12 at 15:05

Let $O_1$ and $O_2$ denote centers of each circle, and $r_1$ and $r_2$ denote their radii. Let $P$ denote the point of intersection you are interested in. We know length of each side of the triangle $\triangle O_1 O_2 P$, hence we can determine its height $h$, i.e. distance from $P$ to the line passing through $O_1$ and $O_2$. It is easiest to do this from the triangle area formulas. Let $d$ denote length of $O_1 O_2$, then $$A(\triangle O_1 O_2 P) = \sqrt{\frac{r_1+r_2+d}{2}\cdot \frac{r_1+r_2-d}{2}\cdot \frac{r_1-r_2+d}{2}\cdot \frac{r_2+d-r_1}{2}} = \frac{1}{2} h d$$ Let $Q$ denote projection of $P$ on $O_1O_2$. Knowing $h$ allows to find length of $O_1Q$ and of $QO_2$ using Pythagorean theorem, allowing to determine coordinates of $Q$, and thus of $P$.
Just set up the two circle equations: $(X-M)^2=r^2$ and follow the instructions.