Posts tagged 'quadrea'

W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants

For any three non-collinear points A_1, A_2, A_3 over a field F (char F != 2) with non-null lines A_1 A_3 and A_2 A_3, the lines A_1 A_3 and A_2 A_3 are perpendicular (s_3 = 1) if and only if Q(A_2, A_3) + Q(A_1, A_3) = Q(A_1, A_2), in which case the hypotenuse midpoint M_3 is equidistant in quadrance (Q_3 / 4) from all three vertices.

W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea

Three planar points A_1, A_2, A_3 with pairwise quadrances Q_1, Q_2, Q_3 are collinear if and only if Archimedes’ symmetric quadratic form A(Q_1, Q_2, Q_3) = (Q_1 + Q_2 + Q_3)^2 - 2(Q_1^2 + Q_2^2 + Q_3^2) vanishes; for any triangle, A(Q_1, Q_2, Q_3) equals 4 times the squared coordinate determinant (16 * Delta^2), defining the triangle’s quadrea.