Posts tagged 'power-of-a-point'

F.8 Chord Intersection

If two chords AB and CD of a circle (O, R) intersect at a point X at distance r = |OX| from the center, then the product of their segments is constant and equals the power of X with respect to the circle: |AX|*|BX| = |CX|*|DX| = R^2 - |OX|^2.

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.