The isodynamic points J, J' of triangle ABC are defined as the intersection points of the Apollonian circles of the sides of the triangle dividing them in the ratio of the two other sides. Naming the sides a=|BC|, b=|CA|, c=|AB|, the three Apollonian circles are: kA = locus of points P: |PB|/|PC| = c/b (and cyclically permuting the symbols), kB = locus of points P: |PC|/|PA| = a/c, kC = locus of points P: |PA|/|PB| = b/a. All three circles pass through the two isodynamic points J, J'.
The figure is reach in relations, examined in the references given below (especially in Apollonian_Circles.html ). Here are listed a few: [1] The line JJ' passes through the circumcenter of ABC (Brocard axis). [2] The centers of the three Apollonian circles lie on the trilinear polar of the symmedian point (Lemoine axis). [3] The three circles are orthogonal to the circumcircle and intersect pairwise under 60 degrees. [4] The inversion on the circumcircle interchanges J, J'. A nice discussion on Isodynamic points and their relation to Fermat's points of a triangle is to be found in [Bottema, section-8].
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