Posts tagged 'chords'

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.13 Compare Triangles in Circles

For a family of triangles ABC inscribed in a fixed circle of radius R sharing a fixed chord AB = c (with central angle 2*phi where sin(phi) = c/(2R)), the area K decomposes into the signed sum of three central isosceles triangles: K = (1/2) c h_c + (1/2) a h_a - (1/2) b h_b.

F.15 Cyclic Quadrilateral

A cyclic quadrilateral WXYZ whose interior angles form the arithmetic progression theta, 2*theta, 3*theta, 4*theta has theta = 36 degrees (W = 36, X = 72, Y = 144, Z = 108) and golden-ratio diagonals |XZ| = 2 sin(36) = sqrt(3 - phi) and |WY| = 2 sin(72) = phi*sqrt(3 - phi) in the unit circle; by contrast, no non-trivial cyclic quadrilateral can have interior angles in geometric progression theta, k*theta, k^2*theta, k^3*theta.