Posts tagged 'apollonius'

F.2 Bilinear Transformations

The canonical bilinear transformation B(x) = (1 + x)/(1 - x) and its inverse B^{-1}(y) = (y - 1)/(y + 1) link the internal division ratio kappa = x/r of a segment AB to the Apollonian distance ratio k = (1 + kappa)/(1 - kappa).

F.12 Coaxial Circles

A hyperbolic pencil of coaxial circles with limiting points L_1 = (-r, 0) and L_2 = (+r, 0) coincides with the Apollonian equipotential circles d_1 / d_2 = k around two point charges Q_1, Q_2, sharing the y-axis as their common radical axis.

F.16 d’Alembert’s Theorem

Given three circles of distinct radii in the plane, their six pairwise internal and external centers of similitude lie by threes on four projective lines L_{+++}, L_{–+}, L_{-+-}, and L_{+–} (the Monge–d’Alembert similitude axes).

F.25 Harmonic Conjugates

Collinear points C and D divide segment AB (of midpoint M and half-length r) internally and externally in the same ratio k = |AC|/|BC| = |AD|/|BD| if and only if |MC|*|MD| = r^2, which occurs if and only if the circle on diameter AB and the Apollonian circle on diameter CD intersect orthogonally.

F.46 Stewart’s Theorem

If a cevian of length f from C divides base c = AB into segments k’ c and k c (with k + k’ = 1), Stewart’s Theorem states f^2 = k a^2 + k’ b^2 - k k’ c^2; at the midpoint k = k’ = 1/2, this yields Apollonius’s median formula f^2 = (1/2)(a^2 + b^2 - c^2/2).

F.51 Apollonian Circles and the Problem of Apollonius

The locus of points P with distance ratio |PA| / |PB| = k > 1 (where |AB| = d) is the Apollonian circle of radius R = d k / (k^2 - 1) centered at distance M’ = d / (k^2 - 1) beyond B, cutting line AB at the internal and external harmonic conjugates C and D such that |A T’|^2 = |AC| |AD| = |AB| |A M’|.