there are a lot of ways to calculate a circle given two points given the formula $(x-k)^2+(y-k)^2=r^2$ on this board. However these are not unique as there are arbitrary many possibilities to draw a circle through two points (with different radius).
I want to add another condition, such that there is only one possible solution. My condition is that one point is the highest point. To sketch my problem, I created an image with R.
As you can see in this specific image, the center is given for $p_0=(0,0)$ (
blue dot, generally unknown), the highest point is given for $p_1=(x_1,y_1)$ (
green dot, highest value, known) and another point $p_2=(x_2,y_2)$ (
red dot, known value) is given as well. The
green lines have the length of the desired radius $r$ (unknown) and the
gray line is known through points $p_1$ and $p_2$.
So to sum up, I know the two values $p_1$ and $p_2$ and I'm searching for the radius $r$. I thought that the
gray line might give me some information utilizing an equal-sided triangle, but so far I had no luck.
So my question is: How can I calculate radius $r$ given two points $p_1$ and $p_2$.