Posts tagged 'harmonic'

F.2 Bilinear Transformations

The canonical bilinear transformation B(x) = (1 + x)/(1 - x) and its inverse B^{-1}(y) = (y - 1)/(y + 1) link the internal division ratio kappa = x/r of a segment AB to the Apollonian distance ratio k = (1 + kappa)/(1 - kappa).

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.7 Ceva and Menelaus

Three cevians AD, BE, CF of triangle ABC concur at a point P if and only if (AF/FB)(BD/DC)(CE/EA) = +1 (Ceva’s Theorem), whereas points D, E, F’ on the sidelines are collinear if and only if (AF’/F’B)(BD/DC)(CE/EA) = -1 (Menelaus’s Theorem), making (A, B; F, F’) a harmonic range.

F.31 Logic Tests

Given collinear points A, B, C, the three Boolean inequality tests T_1 = (|CA| < |AB|), T_2 = (|CB| < |AB|), and T_3 = (|CA| < |CB|) uniquely partition the line AB into Region A (-1 < k < 0), interior segment AB (0 < k < infty), and Region B (-infty < k < -1), determining the sign of k = +/- |AC| / |BC|.

F.37 Euler’s Inradius Parabola

For a fixed circumradius R, Euler’s relation d^2 = R^2 - 2 r R defines a downward-opening parabola r(d) = R/2 - d^2/(2R) = (R - d)(R + d)/(2R) with vertex (0, R/2) at the equilateral triangle and roots at the degenerate limits d = +/- R.

F.39 Pole and Polar with Respect to a Triangle

Given a triangle ABC and a point P with cevian traces A’, B’, C’, the intersections X = BC cap B’C’, Y = CA cap C’A’, and Z = AB cap A’B’ are collinear on the trilinear polar L of the pole P.

F.42 Altitude of a Pythagorean Triple Triangle

In a right triangle with legs a, b and hypotenuse c, the altitude h_c = a b / c satisfies the Inverse Pythagorean Theorem 1/h_c^2 = 1/a^2 + 1/b^2 and divides the hypotenuse into projections u = b^2/c and y = a^2/c with h_c^2 = u y.

F.44 Homothetic Ratios and Crossed Diagonals

Given two parallel uprights of heights h_1 < h_2 separated by horizontal distance d and viewed from their external center of similitude O, the crossed diagonals meet at horizontal distance x = d h_1 / (h_1 + h_2) from the first upright, while O lies at distance y - x = d h_1 / (h_2 - h_1) to its left, forming a harmonic range.

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’|.