Posts tagged 'radical-axis'

F.9 Circle Equations

Pairwise subtraction of Cartesian circle equations (x - u_i)^2 + (y - v_i)^2 = r_i^2 eliminates the quadratic terms x^2 + y^2 to yield linear radical-axis relations, determining both the side Z of a square from three corner distances and the radius of a Sangaku circle tangent to an inscribed square and equilateral triangle.

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.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.