Posts tagged 'euler'

F.32 Markowski’s Theorem

If J, K, L denote the perpendicular sagittas from the side midpoints of triangle ABC to the circumcircle, Markowski’s Theorem states J K L = (1/2) r^2 R = (R^2 - d_IO^2)^2 / (8 R).

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.41 Geometric Problems and Extremal Loci

Among all triangles with longest side c = |AB| = 1, the acute locus inside the vesica piscis above the semicircle on AB has area A = int_0^{sqrt(3)} (sqrt(4 - x^2) - 1) dx - pi/4 = (5 pi - 6 sqrt(3))/12 approx 0.44297 (for c = 2), and maximizing d_IO over acute triangles approaches the degenerate right-isosceles limit C -> (0, 1).