Posts tagged 'equilateral'

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.21 Fermat Point

For any triangle ABC whose largest interior angle is less than 120 degrees, erecting external equilateral triangles BCX_A, CAY_B, ABZ_C on its sides yields concurrent segments AX_A, BY_B, CZ_C that intersect at the Fermat-Torricelli point F = X(13), where <AFB = <BFC = <CFA = 120 degrees and FA + FB + FC is minimized.

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.

F.45 Squares, Subdivided Lattices, and the Japanese Theorem

A square inscribed in a circle of radius R has side s = sqrt(2) R; inside a square of side s, intersecting cevians divide the interior in exact rational proportions, and by Carnot’s Theorem the sum of inradii in any triangulation of a cyclic polygon is invariant.