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A sequence of circles which closes (such as a Steiner chain or the circles inscribed in the arbelos) is called a chain.

In Homogeneous coordinates (x_1,x_2,x_3), the equation of a circle C is a(x_1^2+x_2^2)+2fx_2x_3+2gx_1x_3+cx_3^2=0. The discriminant of this circle is defined as Delta=|a 0 g; ...

A subgroup H of an original group G has elements h_i. Let x be a fixed element of the original group G which is not a member of H. Then the transformation xh_ix^(-1), (i=1, ...

A generalization of the hypergeometric function identity (1) to the generalized hypergeometric function _3F_2(a,b,c;d,e;x). Darling's products are (2) and (3) which reduce to ...

Parallel lines are everywhere equidistant. This postulate is equivalent to the parallel postulate.

In any triangle, if one of the sides is extended, the exterior angle is greater than both the interior and opposite angles.

An algebraic ring which appears in treatments of duality in algebraic geometry. Let A be a local Artinian ring with m subset A its maximal ideal. Then A is a Gorenstein ring ...

If F is a family of more than n bounded closed convex sets in Euclidean n-space R^n, and if every H_n (where H_n is the Helly number) members of F have at least one point in ...

The dilogarithm identity Li_2(-x)=-Li_2(x/(1+x))-1/2[ln(1+x)]^2.

Any complex measure lambda decomposes into an absolutely continuous measure lambda_a and a singular measure lambda_c, with respect to some positive measure mu. This is the ...