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Four or more points P_1, P_2, P_3, P_4, ... which lie on a circle C are said to be concyclic. Three points are trivially concyclic since three noncollinear points determine a ...

Every odd integer n is a prime or the sum of three primes. This problem is closely related to Vinogradov's theorem.

Each point in the convex hull of a set S in R^n is in the convex combination of n+1 or fewer points of S.

product_(k=1)^(n)(1+yq^k) = sum_(m=0)^(n)y^mq^(m(m+1)/2)[n; m]_q (1) = sum_(m=0)^(n)y^mq^(m(m+1)/2)((q)_n)/((q)_m(q)_(n-m)), (2) where [n; m]_q is a q-binomial coefficient.

Any row r and column s of a determinant being selected, if the element common to them be multiplied by its cofactor in the determinant, and every product of another element ...

If the lines joining corresponding points of two directly similar figures are divided proportionally, then the locus of the points of the division will be a figure directly ...

A closed embedded smooth plane curve has at least four vertices, where a vertex is defined as an extremum of curvature.

The identity PVint_(-infty)^inftyF(phi(x))dx=PVint_(-infty)^inftyF(x)dx (1) holds for any integrable function F(x) and phi(x) of the form ...

Given F_1(x,y,z,u,v,w) = 0 (1) F_2(x,y,z,u,v,w) = 0 (2) F_3(x,y,z,u,v,w) = 0, (3) if the determinantof the Jacobian |JF(u,v,w)|=|(partial(F_1,F_2,F_3))/(partial(u,v,w))|!=0, ...

Let V!=(0) be a finite dimensional vector space over the complex numbers, and let A be a linear operator on V. Then V can be expressed as a direct sum of cyclic subspaces.