Let U and V be points on the side line BC of a triangle DeltaABC met by the interior and exterior angle bisectors of angles A. Then the circle with diameter UV is called the A-Apollonian circle. Similarly, construct the B- and C-Apollonian circles (Johnson 1929, pp. 294-299). The Apollonian circles pass through the vertices A, B, and C, and through the two isodynamic points S and S^' (Kimberling 1998, p. 68). The A-Apollonius circle has center with trilinears
alpha:beta:gamma=0:-b:c
(1)
and radius
R_A=(a^2b^2c^2)/((b+c)|b-c|sqrt(-a^4+2a^2b^2-b^4+2a^2c^2-c^4))R,
(2)
where R is the circumradius of the reference triangle.
ApolloniusCirclesRadicalLine
Because the Apollonius circles intersect pairwise in the isodynamic points, they share a common radical line…show more content… The vertices of the D-triangle lie on the respective Apollonius
