Im trying to find radius of given circle below and its center coordinates. Base circle is unit circle with radius 1 as well as coordinates for p1 and p2 are given beforehand
Up to this point I know that $$ |p_1 - c| = r $$
$$ |p_2 - c| = r $$
$$ r^2 + 1 = c^2 $$
But somehow I got stuck to solve and figure out radius and center points of circle. Is there any thing am I missing?
This is how I read the problem: there are 2 arbitrary points M and N with their Cartesian coordinates inside the circle $w$, find the circle $\alpha $ , including points M,N and radius $r$. There are 2 ways of problem's solution: geometrical and analytical.
The circles $w$ and $\alpha $ are called orthogonal. We need to inverse one of a given point relative to $w$, e.g. point$N$, and $N'$ is an inverse image of $N$. The construct of $N'$ is marked green. The circle $\alpha $ including $M$, $N$ and $N'$ is orthogonal to the base circle $w$ in conformance with inversion feature.
$\alpha $: $(x-xp)^2+(y-yp)^2=r^2$
Let us construct 3 equations in Cartesian coordinates having 3 variables $xp, yp, r$:
1) $(xm-xp)^2+(ym-yp)^2=r^2$
2) $(xn-xp)^2+(yn-yp)^2=r^2$
3) $ r^2+1=xp^2+yp^2$
Thus, there are 3 quadratic equations, which can be easily solved for 3 required variables $xp, yp, r$.