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For every positive integer n, there exists a sphere which has exactly n lattice points on its surface. The sphere is given by the equation ...

Consider the average volume of a tetrahedron picked at random inside an octahedron of unit volume. The answer is difficult to compute analytically (Zinani 2003), but the mean ...

The orthic inconic of a triangle is the inconic with inconic parameters x:y:z=cosA:cosB:cosC. (1) It has trilinear equation ...

Two submanifolds X and Y in an ambient space M intersect transversally if, for all p in X intersection Y, TX_p+TY_p={v+w:v in TX_p,w in TY_p}=TM_p, where the addition is in ...

Over a small neighborhood U of a manifold, a vector bundle is spanned by the local sections defined on U. For example, in a coordinate chart U with coordinates (x_1,...,x_n), ...

If two intersections of each pair of three conics S_1, S_2, and S_3 lie on a conic S_0, then the lines joining the other two intersections of each pair are concurrent (Evelyn ...

Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect. In three-dimensional Euclidean space, parallel lines not only fail to ...

pi has decimal expansion given by pi=3.141592653589793238462643383279502884197... (1) (OEIS A000796). The following table summarizes some record computations of the digits of ...

The problem of finding the curve down which a bead placed anywhere will fall to the bottom in the same amount of time. The solution is a cycloid, a fact first discovered and ...

If two perpendicular lines are drawn through the orthocenter H of any triangle, these lines intercept each side (or its extension) in two points (labeled P_(12), P_(12)^', ...