Posts tagged 'euclid'

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.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.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.