Posts tagged 'circumradius'

F.3 Biquadratic Forms in Heron’s Formula

Under the linear substitution X = a/2, W = (b + c)/2, and Y = (c - b)/2, Heron’s area formula reduces to the biquadratic product K = sqrt((W^2 - X^2)(X^2 - Y^2)).

F.4 Carnot’s Theorem

In any triangle ABC, the sum of the signed perpendicular distances from the circumcenter O to the three sides BC, CA, AB equals the sum of the circumradius R and the inradius r: d_a + d_b + d_c = R + 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.