Two tangent circles tangent with central circle

Question

See my problem on Desmos

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There is a circle with radius $R$ at center $(a_1, b_1)$ with the following equation

$(x-a_1)^2+(y-b_1)^2=R^2$

Then a tangent circle with radius $r$ rotated by $\theta_0$ would have the following coordinates.

$a_2 = (R+r)cos(\theta_0)+a_1$

$b_2 = (R+r)sin(\theta_0)+b_1$

But what would be the coordinates for third circle with same radius $r$ that is tangent to both circles?

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What i have tried

let $t$ be the angle from central circle to the arc that ends on two tangent points. (sorry for bad english)

$t= \arcsin(\frac{2r}{R+r})+\theta_0$

the coordinates for third tangent circle with radius $r$ would be

$a_3 = (R+r)cos(t)+a_1$

$b_3 = (R+r)sin(t)+b_1$

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Problem

Although the third circle (green) is tangent to central circle but its not tangent to the other circle (blue). what did i do wrong?

Also i get undefined values if $r$ is bigger than some value.

See my problem on Desmos

enter image description here

Answer 1

You are almost there. You computed the angle $t$ incorrectly. Specifically, it should be (why?) $$t = \theta_0 \pm 2\arcsin\left(\frac{r}{R+r}\right).$$