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When all the points P on one line are related by an isometry to all points P^' on another, the midpoints of the segments PP^' are either distinct and collinear or coincident.

A similarity transformation which transforms each line to a parallel line whose length is a fixed multiple of the length of the original line. The simplest dilation is ...

A topological space that is not connected, i.e., which can be decomposed as the disjoint union of two nonempty open subsets. Equivalently, it can be characterized as a space ...

A ruled surface is called a right conoid if it can be generated by moving a straight line intersecting a fixed straight line such that the lines are always perpendicular ...

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 ...

A theorem stated in 1882 which cannot be derived from Euclid's postulates. Given points a, b, c, and d on a line, if it is known that the points are ordered as (a,b,c) and ...

The point-slope form of a line through the point (x_1,y_1) with slope m in the Cartesian plane is given by y-y_1=m(x-x_1).

Given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter ...

Fano's geometry is a finite geometry attributed to Fano from around the year 1892. This geometry comes with five axioms, namely: 1. There exists at least one line. 2. Every ...

An isoscelizer of an (interior) angle A in a triangle DeltaABC is a line through points I_(AB)I_(AC) where I_(AB) lies on AB and I_(AC) on AC such that DeltaAI_(AB)I_(AC) is ...