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For elliptic curves over the rationals Q, the group of rational points is always finitely generated (i.e., there always exists a finite set of group generators). This theorem ...

The nine-point circle, also called Euler's circle or the Feuerbach circle, is the circle that passes through the perpendicular feet H_A, H_B, and H_C dropped from the ...

In three dimensions, a parallelepiped is a prism whose faces are all parallelograms. Let A, B, and C be the basis vectors defining a three-dimensional parallelepiped. Then ...

Move a point Pi_0 along a line from an initial point to a final point. It traces out a line segment Pi_1. When Pi_1 is translated from an initial position to a final ...

The term perimeter refers either to the curve constituting the boundary of a lamina or else to the length of this boundary. The perimeter of a circle is called the ...

A beautiful general theorem of which the following two statements are special cases (Coxeter and Greitzer 1967, pp. 99-100). 1. If DeltaABC and DeltaA^'B^'C^' are two ...

A quartic algebraic curve also called the peg-top curve and given by the Cartesian equation a^4y^2=b^2x^3(2a-x) (1) and the parametric curves x = a(1+sint) (2) y = ...

A map defined by one or more polynomials. Given a field K, a polynomial map is a map f:K^n->K^m such that for all points (x_1,...,x_n) in K^n, ...

A simple closed curve on a sphere that is not necessarily a great circle but merely intersects as a great circle would (Billera et al. 1999).

The radical line, also called the radical axis, is the locus of points of equal circle power with respect to two nonconcentric circles. By the chordal theorem, it is ...