Posts in category 'Polygons'

F.30 Isosceles Triangles

An isosceles triangle of given semiperimeter s and area K has half-base x = c/2 governed by the cubic 2 s x^3 - s^2 x^2 + K^2 = 0, which under x = y + s/6 reduces to Cardano’s depressed cubic y^3 - (s^2/12) y - (s^3/54 - K^2/(2s)) = 0.

F.33 Euclidean (m, n) Generators

For integers m > n >= 1, Euclid’s quadratic generators a = m^2 - n^2, b = 2 m n, and c = m^2 + n^2 satisfy a^2 + b^2 = c^2 and inradius r = n(m - n).

F.35 Obtuse and Acute Triangles

By Euclid’s Elements II.12 (excess) and II.13 (deficit), the square on side a opposite an obtuse or acute angle satisfies a^2 = b^2 + c^2 +/- 2 c x, where x = |a^2 - (b^2 + c^2)| / (2 c) is the projection of side b onto line c and h = sqrt(b^2 - x^2) is the altitude.

F.40 Regular Polygons and Nested Radicals

For a regular N-gon inscribed in a unit circle (R = 1), doubling the number of sides maps edge length E_{N/2} to E_N = sqrt(2 - sqrt(4 - E_{N/2}^2)), generating closed-form nested quadratic radicals for the 2^k, 3*2^k, 5*2^k, and 15*2^k polygon families.

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).

F.42 Altitude of a Pythagorean Triple Triangle

In a right triangle with legs a, b and hypotenuse c, the altitude h_c = a b / c satisfies the Inverse Pythagorean Theorem 1/h_c^2 = 1/a^2 + 1/b^2 and divides the hypotenuse into projections u = b^2/c and y = a^2/c with h_c^2 = u y.

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.

F.49 Vesica Piscis and Acute-Obtuse Probability

When segment AB of length c is the longest side of triangle ABC, vertex C lies inside the vesica piscis of area c^2 (4 pi - 3 sqrt(3))/6; the semicircle of diameter AB partitions each quadrant of area c^2 (4 pi - 3 sqrt(3))/24 into an obtuse region (O) = c^2 pi / 16 and an acute region (A) = c^2 (5 pi - 6 sqrt(3))/48, giving an exact obtuse-to-acute probability ratio (O)/(A) = 3 pi / (5 pi - 6 sqrt(3)) approx 1.773.

F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction

Given the three interior angles (alpha, beta, gamma) of a triangle normalized to unit area K = 1 (so s = 1/r), the Conway intouch segments x = s - a = r cot(alpha/2), y = s - b = r cot(beta/2), and z = s - c = r cot(gamma/2) determine the side lengths a = y + z, b = z + x, and c = x + y = 2 s - (a + b).