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The hexagon obtained from an arbitrary hexagon by connecting the centroids of each consecutive three sides. This hexagon has equal and parallel sides (Wells 1991). A proof of ...

A circular pattern obtained by superposing parallel equally spaced lines on a set of concentric circles of increasing radii, then coloring the regions in chessboard fashion. ...

In the hyperbolic plane H^2, a pair of lines can be parallel (diverging from one another in one direction and intersecting at an ideal point at infinity in the other), can ...

Let f be a contraction mapping from a closed subset F of a Banach space E into F. Then there exists a unique z in F such that f(z)=z.

If a minimal surface is given by the equation z=f(x,y) and f has continuous first and second partial derivatives for all real x and y, then f is a plane.

Two unit-speed plane curves which have the same curvature differ only by a Euclidean motion.

Let K be a finite complex, let h:|K|->|K| be a continuous map. If Lambda(h)!=0, then h has a fixed point.

Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.

Let f:D(z_0,r)\{z_0}->C be analytic and bounded on a punctured open disk D(z_0,r), then lim_(z->z_0)f(z) exists, and the function defined by f^~:D(z_0,r)->C f^~(z)={f(z) for ...

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and n n-polygonal numbers. ...