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There are two theorems commonly known as Feuerbach's theorem. The first states that circle which passes through the feet of the perpendiculars dropped from the polygon ...

There are three theorems related to pedal circles that go under the collective title of the Fontené theorems. The first Fontené theorem lets DeltaABC be a triangle and P an ...

The envelope of the plane lx+my+nz=c, (1) where c is the speed of propagation of a wave in the direction (l,m,n) (i.e., l, m, and n are the direction cosines) is known as the ...

The generalized law of sines applies to a simplex in space of any dimension with constant Gaussian curvature. Let us work up to that. Initially in two-dimensional space, we ...

In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a group G acts on a set X (this process is called a group action), ...

There are a number of mathematical curves that produced heart shapes, some of which are illustrated above. A "zeroth" curve is a rotated cardioid (whose name means ...

The Laplacian for a scalar function phi is a scalar differential operator defined by (1) where the h_i are the scale factors of the coordinate system (Weinberg 1972, p. 109; ...

A 24-dimensional Euclidean lattice. An automorphism of the Leech lattice modulo a center of two leads to the Conway group Co_1. Stabilization of the one- and two-dimensional ...

Roughly speaking, a tangent vector is an infinitesimal displacement at a specific point on a manifold. The set of tangent vectors at a point P forms a vector space called the ...

Let T_1 be the point at which the J_1-excircle meets the side A_2A_3 of a triangle DeltaA_1A_2A_3, and define T_2 and T_3 similarly. Then the lines A_1T_1, A_2T_2, and A_3T_3 ...