|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
points, 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 are
concurrent, and to prove
interesting inversive properties. It also avoids the problem with an
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.
For a triangle ABC, the
A-Apollonian is the member of the Apollonian family A(B,C) through A.
The Apollonians of ΔABC meet at two
points inverse with respect to the circumcircle.
They meet at ∞ if and
only if ΔABC is equilateral.
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.
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.