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AB is tangent to the inner circle, consider the trigonometric circle.

Knowing the radius of both circles, is there a relation between those 2 point's coordinates ?

Their coordinate being $A = (R \cos(\alpha)$, $R \sin(\alpha))$ and $B = (R \cos(\beta)$, $R \sin(\beta))$, $\alpha$ and $\beta$ being the angle formed by the arcs $A$ and $B$.

Meaning that for example if I know $A$, could I calculate $B$, or vice versa ?

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2 Answers 2

$B$ is always 90 degrees, and $\sin(A)=r/R$

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I edited my question for more clarity –  jokoon Aug 28 '12 at 21:43

I'll try to be rather explicit enough because I can't make a sketch. Let $R$ and $r$ be the radii of the big and the small circles respectively -- $R>r$. Let $O$ be the centre of the two circles. Let $\theta$ be the angle $\widehat{AOB}$. Notice that $tan\cos=\frac{r}{R}$, hence $\theta=\arccos\frac{r}{R}$. If we assign the position $\theta=0$ at the point $B$ (in polar coordinates) then the point $A$ will have polar coordinates $(R,-\theta)$.

If you like you may convert these coordinates to some Cartesian system. You may also have some offset in the sense that $B$ is placed at some angle $\phi$, so the angle of $A$ becomes $\phi-\theta$ etc.

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isn't it rather $cos \theta = \frac{r}{R}$ ? –  jokoon Aug 29 '12 at 7:44
    
@jokoon thanks! –  Pantelis Sopasakis Aug 29 '12 at 8:27

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