I would like to calculate the maximum diameter of a circle that is inscribed by the intersection of 3 other circles. The 3 circles have known diameters and center point coordinates. See illustration here.
Where the red, green, and blue circles are given and the black is to be calculated.
It may also be the case that the inscribed circle is tangent to only two of the circles as shown in this illustration.
Is this already a known mathematical example with a name that I can look up? How would I approach solving this? Any help is greatly appreciated.
Thanks,
Dillon
You can start with inscribing circle into the intersection of two circles:
and the use this configuration as a building block.
For given intersection of two circles $\bigcirc(A,r_a)$ and $\bigcirc(B,r_b)$, $|AB|=d$, the maximal radius of the inscribed circle
\begin{align} r_{\max}&=\tfrac12\,(r_a+r_b-d) . \end{align}
Considering $A$ at the origin, the coordinates of the center $O$ of inscribed circle $\bigcirc(O,r)$
\begin{align} O_x&= d-\frac{(r_b-r)\Big(d^2+(r_b-r)^2-(r_a-r)^2\Big)}{2d(r_b-r)} ,\\ O_y&= (r_b-r)\sqrt{1-\Big(\frac{d^2+(r_b-r)^2-(r_a-r)^2}{2d(r_b-r)}\Big)^2} . \end{align}
The blue line on the diagram represents the locus of the centers of all possible inscribed circles.
Similarly, the other pair of circles, say, $\bigcirc(B,r_b)$ and $\bigcirc(C,r_c)$ can be added to the diagram (with proper rotation), and the intersection of the loci curves will give a center of the circle, inscribed in the intersection of all the three circles, $\bigcirc(A,r_a),\,\bigcirc(B,r_b)$ and $\bigcirc(C,r_c)$.
For example, like this: