Finding a circle tangent to three given circles

Question

Given the position and the radius of the violet and blue circles. I need to find the position of the pink circle which should touch all previous circles.

Any ideas?

Answer 1

If you call the radii of the circles $R_a$, $R_b$ and $R_c$ and use the Pythagorean theorem one finds $$ (R_a + R_b)^2 + (R_a + R_c)^2 = (R_b + R_c)^2 $$ Solving for $R_c$ yields: $$ R_c = \frac{R_a(R_a+R_b)}{R_a-R_b}$$ If $R_a \neq R_b$. Filling in $R_a = 2$ and $R_b = 4$ (which I eyeballed from the picture) gives $R_c = 6$