Posts tagged 'barycentric'

F.6 Centroid

The three medians AM_A, BM_B, CM_C joining the vertices of triangle ABC to the midpoints of the opposite sides concur at the centroid G = X(2), which divides each median in a 2 : 1 ratio and has homogeneous barycentric coordinates 1 : 1 : 1.

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.11 Circumcenter

The perpendicular bisectors of the three sides of triangle ABC concur at the circumcenter O = X(3), equidistant from A, B, C by the circumradius R = abc/(4K), with homogeneous barycentric coordinates a^2 S_A : b^2 S_B : c^2 S_C.

F.14 Conway Notation

In John H. Conway’s triangle notation, setting S = 2K (twice the area of triangle ABC), S_omega = (a^2 + b^2 + c^2)/2, and S_A = (b^2 + c^2 - a^2)/2, S_B = (c^2 + a^2 - b^2)/2, S_C = (a^2 + b^2 - c^2)/2 reduces trigonometric and barycentric identities to polynomial expressions satisfying S_B S_C + S_C S_A + S_A S_B = S^2.

F.28 Incenter

The three interior angle bisectors of triangle ABC concur at the incenter I = X(1) with homogeneous barycentric coordinates a : b : c, inradius r = K/s satisfying 1/r = 1/h_a + 1/h_b + 1/h_c, and Euler’s distance d_{IO}^2 = R(R - 2r).

F.52 Homogeneous Barycentric Coordinates

Every triangle center X(n) has homogeneous barycentric coordinates (u : v : w) normalizable to absolute weights lambda_A + lambda_B + lambda_C = 1 such that X = lambda_A A + lambda_B B + lambda_C C; for the nine-point center N = X(5), lambda_A = 1/4 + (b^2 - c^2)(a^2 - (b^2 + c^2)) / (32 K^2).

F.54 HP-35s Triangle Program and Coordinate Scripts

In the canonical frame A = (0, 0), B = (M, 0), C = (I, J), the forward and inverse barycentric-Cartesian transformations BARY 2 CAR and CAR 2 BARY are given by P = M U + I V, Q = J V and V = Q / J, U = (P - I V) / M, T = 1 - (U + V).