How to calculate the boundary points and center of a circle fitting inside a crescent?

Question

Given are two intersecting circles $A, B$ with radius $r_A > r_B$ and center $M_A, M_B$. A third circle $C$ with radius $r_C$ and center $M_C$ fits in the non intersecting part of circle $B$ and touches the two circles $A, B$ at the points $P_{CA}, P_{CB}$. What are the results for $M_C, P_{CA}, P_{CB}$ for given $A, B, r_C$?

Not part of question, only thougts on solving: Does the path for $M_C$ describe a part of an ellipse? This could be the first step to the solution.

Answer 1

We may assume $M_A=(0,0)$, $M_B=(d,0)$, $\>d>0$, and let $a$, $b$ be the radii of the two circles $A$ and $B$. Let $M_C=(u,v)$ be the center of a touching circle and $r$ its radius. The conditions $|M_C M_A|= a+r$ and $|M_C M_B|=b-r$ enforce $$u^2+v^2=(a+r)^2,\qquad(u-d)^2+v^2=(b-r)^2\ .\tag{1}$$ Subtracting the second of these equations from the first gives $$2ud -d^2=(a+b)(a-b+2r)\ .$$ This determines $u=u(r)$, and the first equation $(1)$ then gives $v=\pm v(r)$, whereby $0\leq r\leq{1\over2}(d+b-a)$, by inspection of the figure.

Answer 2

I have put a Geogebra sheet at https://www.geogebra.org/m/qnFkgnev. I am going to write the 4 equations needed to solve for the smallest circle. A convention that I am using: If I have fixed the value of a point, then it is addressed as $(x(P),y(P))$, whereas, if I have the point as an unknown, it is addressed as $(P_x,P_y)$ In the Geogebra sheet, you can move point C about but the browser is slow. So, click, hold down and move cursor to a new spot on the circle, release. Wait a few seconds and hopefully all will recalculate/redraw. My 4 unknowns are $D_x,D_y,E_x,E_y$

1. Unknown point D, is on the line between B & C. m is the slope of that line. $$m=\left(\frac{y(B)-y(C)}{x(B)-x(C)}\right)$$ The equation: $$D_y=y(C)+m(D_x-x(C))$$

2. The radius of the small circle is the same between C & D and between E & D. $$|C-D|=|E-D|$$

3. Point E has to lie on circle A. $$E_x^2+E_y^2-3E_x-3E_y=4.5$$ (I think this circle is hard coded in the geogebra sheet, hence the numerical values shown in this equation.)

4. Point D must lie on a line from A through E. $$m_1=\left(\frac{E_y-y(A)}{E_x-x(A)}\right)$$ and the equation: $$D_y=E_y+m_1(D_x-E_x)$$

Once these were solved (by Geogebra CAS), I could move the fixed point C about and a new point D was generated. I saved these using $SetValue(D_n,D)$. Next I chose 3 of the D values and put a circle through them. Surprise (or not), that circle does not quite go through all of the D values. i.e. the locus of points in question do not quite make the arc of a circle. Is it an ellipse? Maybe, but that certainly isn't proven yet.