Posts tagged 'circles'

F.5 Centers of Similitude

Given two circles (O_1, r_1) and (O_2, r_2) separated by center distance d = |O_1 O_2| with r_2 > r_1, their common internal and external tangents concur at the internal and external centers of similitude S_IN and S_EX, which divide O_1 O_2 harmonically in the ratio r_1 : r_2.

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.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.10 Circles and Geometric Mean

In a semicircle on diameter AB = m + n, the perpendicular altitude h = CH from a point C on the circumference to AB is the geometric mean sqrt(m*n) of the segments AH = m and HB = n, while the radius OC = (m + n)/2 is their arithmetic mean.

F.24 Gothic Arch

In a square ABCD of side s with pointed Gothic arch arcs of radius s centered at A(0, 0) and B(s, 0) meeting at apex E(s/2, (sqrt(3)/2)s), an inscribed tracery circle centered at P(s/2, Q) of radius r = s - p satisfies p*s = P_x H_x + P_y H_y at its tangency point H.

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.