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A finite-dimensional Lie algebra all of whose elements are ad-nilpotent is itself a nilpotent Lie algebra.

If three skew lines all meet three other skew lines, any transversal to the first set of three meets any transversal to the second set of three.

When a point P moves along a line through the circumcenter of a given triangle Delta, the pedal circle of P with respect to Delta passes through a fixed point (the Griffiths ...

If two pairs of opposite polygon vertices of a complete quadrilateral are pairs of harmonic conjugate points, then the third pair of opposite polygon vertices is likewise a ...

A necessary and sufficient condition for a measure which is quasi-invariant under a transformation to be equivalent to an invariant probability measure is that the ...

Of all convex n-gons of a given perimeter, the one which maximizes area is the regular n-gon.

A necessary and sufficient condition for a curve to be a helix is that the ratio of curvature to torsion be constant.

If every component L of X/O_(p^')(X) satisfies the "Schreler property," then L_(p^')(Y)<=L_(p^')(X) for every p-local subgroup Y of X, where L_(p^') is the p-layer.

An important result in ergodic theory. It states that any two "Bernoulli schemes" with the same measure-theoretic entropy are measure-theoretically isomorphic.

Each row and each column in the group multiplication table lists each of the group elements once and only once. From this, it follows that no two elements may be in the ...