Isodynamic points

[Arne]

Can someone tell me some useful things about the isodynamic points of a triangle?

[darij grinberg]

Hmm... try searching the web... Well, Clark Kimberling has a definition with a figure... then there is a nice page about them by Peter Woo... and somewhere else you might find these nice properties: If ABC is a triangle, then the A-Apollonian circle is the locus of all points P such that BP : CP = c : b. And hence, the only points P satisfying the property AP : BP : CP = (1/a) : (1/b) : (1/c) are the points of intersection of these three Apollonian circles, i. e. the two isodynamic points of triangle ABC. Or you could find out that the two isodynamic points are mutually inverse with respect to the circumcircle of triangle ABC; hence, the line joining them passes through the circumcenter O of triangle ABC. This line also passes through the symmedian point K of triangle ABC. The two isodynamic points are the isogonal conjugates of the two Fermat points. Finally, the pedal triangle of a point P with respect to the triangle ABC is equilateral if and only if P is one of the two isodynamic points of triangle ABC. This is already quite a number of results, isn't it?

BTW, I was rather unprecise when I wrote that they're so popular in IMOs. Actually the two most popular geometric themes are isodynamic points and pedal triangles. But this all is interconnected, since the pedal triangles of the isodynamic points are equilateral

Darij

[radko]

Do you know where the name "isodynamic" comes from?
The word "Isodynamic" suggests that some forces are equal. Which ones? This must be related to the property of distances to the vertices.
Does someone know a reference?

Thanks!