Posts in category 'Circle Inversion'

F.32 Markowski’s Theorem

If J, K, L denote the perpendicular sagittas from the side midpoints of triangle ABC to the circumcircle, Markowski’s Theorem states J K L = (1/2) r^2 R = (R^2 - d_IO^2)^2 / (8 R).

F.43 Radical Axes and Radical Center

The radical axis of two circles of radii r_1, r_2 with center separation d cuts the line of centers at x = d/2 + (r_1^2 - r_2^2)/(2 d) from O_1, and the three radical axes of the vertex-altitude circles Gamma_A(h_A), Gamma_B(h_B), Gamma_C(h_C) concur at the radical center W.

F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety

If lines parallel to the sides of triangle ABC are drawn tangent to the incircle, the three corner subtriangles are homothetic to ABC with scale factors x/s, y/s, z/s (where x = s - a, y = s - b, z = s - c), so their inradii satisfy r_A + r_B + r_C = r.

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’|.