Posts in category 'Special Triangles & Sacred Proportions'

F.18 Equilateral Triangle

In an equilateral triangle of side a, the incenter, centroid, circumcenter, and orthocenter coincide, with altitude h_A = (sqrt(3)/2)a = 3r and R = 2r; moreover, points A’, B’, C’ placed along the angle bisectors AI, BI, CI of any triangle ABC at distances A’I = (2r/sqrt(3)) sin(alpha/2 + pi/6) form an equilateral triangle.

F.23 Golden Triangles

In the acute golden isosceles triangle ABC with base AB = 1, legs AC = BC = phi, and angles 72-72-36 degrees, bisecting base angle B meets AC at D such that CD = 1, DA = phi - 1 = 1/phi, and BD = AB = 1, partitioning ABC into a smaller golden triangle BCD and a golden gnomon ABD.

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.