Posts tagged 'harmonic-conjugates'

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.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.