of center A and radius r is the loci of points whose distance from A is given by r. The point A is the center of the circumference.

any 3 points A, B and C are colinear, if and only if, there exists an straight line that contain all 3 points.

is a supposed valid mathematical result, for wich a demonstration is still unknown See theorem and demonstration.

a set A is convex if and only if, for any (a¹,a²,l) in A x A x [0,1], l a¹ + (1-l) a² is also an element of A.
This means that the segment with ending points in any element of A lies entirely in the set A.

definida pelos pontos A e B é um sub-conjunto contíguo (curva) da circunferência delimitado por ambos os pontos. Note que deste modo, estão definidas duas cordas.

is a transformation (function) that preserves distances (in metric spaces). Threrefore, if f:D -> I is an isometry, then any pair a and b in the domain D, ||a-b|| = ||f(a)-f(b)||, where ||x-y|| is the distance between x and y.
This means that , ambos no domínio D

P is the midpoint of AB, then P is in AB and ||P-A||=||P-B||.
Therefore, the midpoint of A and B, is the point P in the segment AB, whose distance of to both ends is equals.

m of the points A and B is the straight line passing by the midpoint, orthogonal to the segment AB. Note that this is equivalente to the following definition: m is the loci of points whose distances from A and B is equals, ie, m := { P: ||P-A||=||P-B||}.

a function is called recursive or recurrent when in its definition there exists an invocation to itself.
The most famous recursive function is the factorial function: The recurence in the factorial function is characterized by the invocation of itself (as in the case of recursive script) in the "right side" of its definition (in the above example, the recurence is defined by the invocation fat(n-1), as highlighted with the underline).