In the image below, the two smallest circles and the chord are given to me.
What I would like to calculate is the radius of the largest circle, such that it includes the endpoints of the chord and is tangential to the smallest circle. Is there a way to calculate this symbolically?
The following is what I know:
- The center of medium circle is at the origin $(0,0)$
- The radius of the medium circle is $R_m$, and the small circle is $R_s$
- Chord endpoints are given at $(x_1, y_1)$ and $(x_2, y_2)$
- The center of the smallest circle lies on the medium circle at point $(S_x, S_y)$
- Both intersections of the medium and small circles occur within the chord
- The center point of the largest circle can be written as $(L_x, L_y)$, where $L_x=\frac12(x_1+x_2)t$, and $L_y=\frac12(y_1+y_2)t$, for some $t > 1$.
I need to compute the center point of the large circle $(L_x, L_y)$ given this information. I have created the following system of two equations with two unknowns ($R_I$ and $t$) given this information, but can't figure out how to solve them:
Any help is greatly appreciated.