odds and ends - the isodynamic points

 In Kimberling's encyclopedia, the isodynamic points of a triangle ABC are defined as intersections of three circles. From vertex A, we draw the internal and external bisectors, meeting BC at U,V. We draw the circle on UV as diameter. Similarly, we draw circles starting from B and C. These three meet in twopoints, the isodynamic points. Kimberling points out that the circles are the Apollonian circles of ΔABC. Here, we define the points in terms of the Apollonian circles. It is then easier to establish that they areconcurrent, and to prove interesting inversive properties. It also avoids the problem with an isosceles triangle, namely that one external bisector is parallel to the opposite side, so V=∞ and the "circle" on"diameter" UV must be taken as the perpendicular bisector of the side. definitionFor a triangle ABC, the A-Apollonian is the member of the Apollonian family A(B,C) through A. Theorem A1The Apollonians of ΔABC meet at two points inverse with respect to the circumcircle.They meet at ∞ if and only if ΔABC is equilateral. proof Observe that an inversive map takes apollonian families to apollonian families. We choose an inversive map t taking A to ∞ and B,C to B',C', say. The B- and C-apollonian circles map to circles through B', centre C' and through C', centre B' respectively. By symmetry, these meet in two points on the perpendicular bisector of B'C', and these points are equidistant from B'C'. Thus, they are inverse with respect to B'C'. But the perpendicular bisector is the member of A(B',C') through ∞, i.e. the ∞-apollonian. Thus, the three apollonians meet in two points. Also, the circumcircle of ΔABC maps to the line B'C'. Applying t-1, we have the first part. In general, the A-apollonian is a line if and only if the perpendicular bisector of BC passes through A, i.e. the angles at B and C are equal. It follows at once that all the apollonians are lines - so intersect at ∞ - if and only if ΔABC is equilateral. definition The isodynamic points of a triangle are the points of concurrence of its apollonians. The first isodynamic point is that inside the circumcircle, the second that outside.