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Let C be a curve, let O be a fixed point (the pole), and let O^' be a second fixed point. Let P and P^' be points on a line through O meeting C at Q such that P^'Q=QP=QO^'. ...

Four circles may be drawn through an arbitrary point P on a torus. The first two circles are obvious: one is in the plane of the torus and the second perpendicular to it. The ...

Given an arbitrary planar quadrilateral, place a square outwardly on each side, and connect the centers of opposite squares. Then van Aubel's theorem states that the two ...

A general quadratic Diophantine equation in two variables x and y is given by ax^2+cy^2=k, (1) where a, c, and k are specified (positive or negative) integers and x and y are ...

There exist infinitely many odd integers k such that k·2^n-1 is composite for every n>=1. Numbers k with this property are called Riesel numbers, while analogous numbers with ...

The crossed trough is the surface z=x^2y^2. (1) The coefficients of its first fundamental form are E = 1+4x^2y^4 (2) F = 4x^3y^3 (3) G = 1+4x^4y^2 (4) and of the second ...

Hilbert's problems are a set of (originally) unsolved problems in mathematics proposed by Hilbert. Of the 23 total appearing in the printed address, ten were actually ...

The number two (2) is the second positive integer and the first prime number. It is even, and is the only even prime (the primes other than 2 are called the odd primes). The ...

Define the first Brocard point as the interior point Omega of a triangle for which the angles ∠OmegaAB, ∠OmegaBC, and ∠OmegaCA are equal to an angle omega. Similarly, define ...

The Brocard circle, also known as the seven-point circle, is the circle having the line segment connecting the circumcenter O and symmedian point K of a triangle DeltaABC as ...