
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?

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?
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 $P$.
Here is a nice example: http://www.analyzemath.com/CircleEq/circle_intersection.html
Just set up the two circle equations: $(X-M)^2=r^2$ and follow the instructions.