Proving the existence of first isodynamic point.

Question

Info: A $P$-Appolonius circle of two points $A,B$ is the locus of point P such that $\frac{AP}{BP}$ is a constant $AP\neq PB$. This locus is a circle and is not difficult to prove.

Question: Given a scalene triangle $ABC$, show that the $A-$,$B-$,$C-$appolonius circles concur at a point inside $ABC$.

I was trying radical axis theorem using circumcircle and the appolonius circles, but cant proceed, please help. Thank You.

Answer 1

Let $X$ be the intersection point of the $B$- and $C$-Apollonius circles of $\triangle ABC$ that lies inside $\triangle ABC$. Then $$ \frac{AX}{CX} = \frac{AB}{CB} \quad \mbox{and} \quad \frac{AX}{BX} = \frac{AC}{BC}, $$ so ...