Posts tagged 'inversion'

F.26 Harmonic Conjugate Inversion

For a circle Gamma of center M and radius r on diameter AB, internal point C at x = |MC| and its harmonic conjugate D at y = |MD| are inverses satisfying x*y = r^2, while the tangent DT = T forms a parallelogram M D M_T T whose long diagonal z satisfies z^2 = y^2 + 3*r^2 by Stewart’s Theorem.

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’|.