Hello,

I was wondering if you could elaborate on the proof for the circles of Apollonius.
(http://www.cut-the-knot.org/Curriculum/Geometry/LocusCircle.shtml)

What argument did you have in mind behind the statement "For any P on the circle, the internal and external bisectors of angle APB pass through (the fixed points) M and N."?

I came up with one but I am guessing yours is simpler. Here's mine:

Take P on the circle, bisect the angle APB. If the bisector intersects the line AB at M' different from M, then assume WLOG M' is closer to B than is M. Then we get N' such that N'B/N'A = M'B/M'A and one can show that N' is closer to B than N is. Running the construction again we get a new circle with diameter M'N' properly within circle MN, but having intersection P, which is impossible.

Is there a simpler one reasoning from similar triangles and the like?

Thanks,
Monroe