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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.

Diagram

F is derived from:

  1. finding the tangents between 1) $C_R$ and $C_G$ and 2) $C_R$ and $C_B$,
  2. offsetting each tangent by 1) $r_G$ and 2) $r_B$
  3. 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)

meetamit
  • 165

1 Answers1

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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.

enter image description here

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, $h = - (RP + PJ)$.

Mick
  • 17,141