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A polygonal number of the form O_n=n(3n-2). The first few are 1, 8, 21, 40, 65, 96, 133, 176, ... (OEIS A000567). The generating function for the octagonal numbers is ...

The vertex of an isosceles triangle having angle different from the two equal angles is called the apex of the isosceles triangle. The common polygon vertex at the top of a ...

Given a Schwarz triangle (p q r), replacing each polygon vertex with its antipodes gives the three colunar spherical triangles (p q^' r^'),(p^' q r^'),(p^' q^' r), (1) where ...

The dual of a regular tessellation is formed by taking the center of each polygon as a vertex and joining the centers of adjacent polygons. The triangular and hexagonal ...

Define A^' to be the point (other than the polygon vertex A) where the triangle median through A meets the circumcircle of ABC, and define B^' and C^' similarly. Then the ...

Pick a point O in the interior of a quadrilateral which is not a parallelogram. Join this point to each of the four vertices, then the locus of points O for which the sum of ...

A mensuration formula is simply a formula for computing the length-related properties of an object (such as area, circumradius, etc., of a polygon) based on other known ...

The problem of polygon intersection seeks to determine if two polygons intersect and, if so, possibly determine their intersection. For example, the intersection of the two ...

If, in a plane or spherical convex polygon ABCDEFG, all of whose sides AB, BC, CD, ..., FG (with the exception of AG) have fixed lengths, one simultaneously increases ...

Two polygons are congruent by dissection iff they have the same area. In particular, any polygon is congruent by dissection to a square of the same area. Laczkovich (1988) ...