# Inscribed circle: find distance to circumscribed circle

Let a circle with center $b$ and radius $r$ be contained in a circle with center $a$ and radius $R$.

Given a point $c$ on the small circle, find its distance to the greater circle.
That is find the length of $cd$.
Assume
- the length $ab$, $r$ and $R$ are known,
- the $\mathbb R^2$ coordinates of $a$, $b$ and $c$ are known.

Using the dot product, I can find the angle $abc$.
But, then the cosine law does not uniquely determined the length $bd$. $$r + cd = \left(ab\right) \cos abc \pm \sqrt{R^2 - \left(ab\right)^2 \sin^2 abc}.$$

Is there a way to uniquely determine $cd$?

• If I understand correctly, I draw the tangent $t$ at the point $c$. Then I draw a tangent $T$ of the great circle parallel to $t$. Let say $T$ is on the side of the point $d$. Then the distance between the tangencial contact points is the minimum distance from the point $c$ to the great circle. I don't see how this resolve the uniqueness of the length $cd$.
• Also the tangent $T$ does not pass through the point $d$. And the distance $cd$ seems shorter than the distance $tT$ (at the tangencial contact point).