let C1, C2 be two circles that intersect each other so we get two Points P1, P2 that lie on both circles.
Now we construct a circle $C_i$ with center $M_i$ so that this circle tangents C1 and C2 from inside.
How can I mathematically find the locus of all the $M_i$ for those circles?
I thought about something like:
the distance from every $M_i$ to the Point where $C_i$ touches C1 has the same length as the distance from $M_i$ to the Point where the same $C_i$ touches C2…?
Best Answer
... Thus the difference of the distances to the circles' centers is constant. The locus is therefore part of a hyperbola having the centers of the circles as focii. In fact, it is subarcs of a branch on this hyperbola.
Of course, points $P_i$ are points of these subarcs.