Pardon the amateurish notation and what's probably a very simple puzzle for most on this site. Still, the solution eludes me....
I have 3 circles $C_R, C_G, C_B$ as shown in the diagram. I know the $x$ and $y$ coords of each circle's center, as well as their respective radii, $r_R, r_G, r_B$.
I am looking for the formula to compute the x and y coords of the purple point F in terms of the 3 circles' x, y and r.
F is derived from:
- finding the tangents between 1) $C_R$ and $C_G$ and 2) $C_R$ and $C_B$,
- offsetting each tangent by 1) $r_G$ and 2) $r_B$
- computing the intersection F of the two offset lines
(A potentially simplifying assumption here is that $C_G$ and $C_B$ are horizontally aligned, ie their centers always have the same x coordinate)
WLOG, we can assume that R is at (0, 0). I also assume that the equation of $L$, the external tangent to $C_R$ and $C_G$ can be found.
The green dotted circle (called $C_H$) is the translated image of $C_G$ so that it touches $C_R$ internally at $S$. The same can be done to $C_B$.
Clearly, $F = (h, y_B)$, where $y_B$ is known and $h$ has to be determined.
$RM = y_M$, a known quantity which is given by $M(0, y_M)$, the point of intersection of $L$ and $x = 0$. Note that $RS = r_R$ is also a known quantity. Then, $\theta$ in $\triangle RSM$ can be found via $\cos \theta = \dfrac {r_R}{y_M}$.
$RH (= r_R - r_G)$, $RP$ and $PJ$ can also be subsequently found. Finally, $h = - (RP + PJ)$.