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A formula relating the number of polyhedron vertices V, faces F, and polyhedron edges E of a simply connected (i.e., genus 0) polyhedron (or polygon). It was discovered ...

Given a function f(x)=f_0(x), write f_1=f^'(x) and define the Sturm functions by f_n(x)=-{f_(n-2)(x)-f_(n-1)(x)[(f_(n-2)(x))/(f_(n-1)(x))]}, (1) where [P(x)/Q(x)] is a ...

The Johnson midpoint is the point of concurrence of the line segments joining the vertices of a reference triangle with the centers of a certain set of circles (that resemble ...

There are two nonintersecting circles that are tangent to all three Lucas circles. (These are therefore the Soddy circles of the Lucas central triangle.) The inner one, ...

The resistance distance between vertices i and j of a graph G is defined as the effective resistance between the two vertices (as when a battery is attached across them) when ...

An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r_1 and r_2 from two fixed points F_1 and F_2 (the foci) separated by a distance ...

If an n-sided Poncelet transverse constructed for two given conic sections is closed for one point of origin, it is closed for any position of the point of origin. ...

The constant pi, denoted pi, is a real number defined as the ratio of a circle's circumference C to its diameter d=2r, pi = C/d (1) = C/(2r) (2) pi has decimal expansion ...

A system of circles obtained by multiplying each radius in a coaxal system by a constant. The Tucker circles are a coaxaloid system (Johnson 1929, p. 277).

There are four completely different definitions of the so-called Apollonius circles: 1. The set of all points whose distances from two fixed points are in a constant ratio ...