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The triangle with edge lengths 3, 4, and 5 is the right triangle with smallest possible integer lengths and corresponds to the Pythagorean triple (3,4,5) where the legs have ...

Affine functions represent vector-valued functions of the form f(x_1,...,x_n)=A_1x_1+...+A_nx_n+b. The coefficients can be scalars or dense or sparse matrices. The constant ...

Given a point P and a line AB, draw the perpendicular through P and call it PC. Let PD be any other line from P which meets CB in D. In a hyperbolic geometry, as D moves off ...

The region lying between two concentric circles. The area of the annulus formed by two circles of radii a and b (with a>b) is A_(annulus)=pi(a^2-b^2). The annulus is ...

Two points which are collinear with respect to a similitude center but are not homologous points. Four interesting theorems from Johnson (1929) follow. 1. Two pairs of ...

Enclose a sphere in a cylinder and cut out a spherical segment by slicing twice perpendicularly to the cylinder's axis. Then the lateral surface area of the spherical segment ...

A point B is said to lie between points A and C (where A, B, and C are distinct collinear points) if AB+BC=AC. A number of Euclid's proofs depend on the idea of betweenness ...

The bigyrate diminished rhombicosidodecahedron is a convex equilateral solid that is Johnson solid J_(79). The unit bigyrate diminished rhombicosidodecahedron has volume ...

A coordinate system obtained by inversion of the bicyclide coordinates. They are given by the transformation equations x = Lambda/(aUpsilon)snmudnnucospsi (1) y = ...

The term "Cartesian" is used to refer to anything that derives from René Descartes' conception of geometry (1637), which is based on the representation of points in the plane ...