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The total power of a triangle is defined by P=1/2(a_1^2+a_2^2+a_3^2), (1) where a_i are the side lengths, and the "partial power" is defined by p_1=1/2(a_2^2+a_3^2-a_1^2). ...

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 ...

A graph is claw-free iff it does not contain the complete bipartite graph K_(1,3) (known as the "claw graph"; illustrated above) as a forbidden induced subgraph. The line ...

The cotangent function cotz is the function defined by cotz = 1/(tanz) (1) = (i(e^(iz)+e^(-iz)))/(e^(iz)-e^(-iz)) (2) = (i(e^(2iz)+1))/(e^(2iz)-1), (3) where tanz is the ...

A factorial prime is a prime number of the form n!+/-1, where n! is a factorial. n!-1 is prime for n=3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, ...

Let Pi be a permutation of n elements, and let alpha_i be the number of permutation cycles of length i in this permutation. Picking Pi at random, it turns out that ...

A quartic symmetric graph on 30 nodes and 60 edges corresponding to the skeleton of the Archimdean icosidodecahedron, great dodecahemidodecahedron, great icosidodecahedron, ...

The Jacobi triple product is the beautiful identity product_(n=1)^infty(1-x^(2n))(1+x^(2n-1)z^2)(1+(x^(2n-1))/(z^2))=sum_(m=-infty)^inftyx^(m^2)z^(2m). (1) In terms of the ...

The Kepler-Poinsot polyhedra are four regular polyhedra which, unlike the Platonic solids, contain intersecting facial planes. In addition, two of the four Kepler-Poinsot ...

The Kiepert hyperbola is a hyperbola and triangle conic that is related to the solution of Lemoine's problem and its generalization to isosceles triangles constructed on the ...