Lines {line} are straight and have no endpoint.
Line segments can connect {join} two points.
Dividing line segment internally or externally can divide second line segment in same proportions {proportional division}. Put second-segment end on first-segment end. Then draw straight line between other endpoints. Then draw line parallel to straight line at first-line-segment dividing point.
Figures {rectilinear figure, line} can have only straight lines.
Line-segment multiples can have greater length than any line segment {Archimedes axiom, line} {axiom of Archimedes, line}.
Plane-geometry axioms {Euclid's twelfth axiom} {Euclid twelfth axiom} can claim that parallel lines exist.
One and only one straight line through a point not on another straight line does not intersect second straight line {parallel postulate}|.
Euclid's twelfth axiom can be simpler {Playfair's axiom} {Playfair axiom}. Through any point, only one line can be parallel to fixed line. Given straight line and exterior point, only one straight line is parallel to the line.
Three lines can meet at angles of 120 degrees {isometric axes}.
Lines can intersect coordinate planes at one point {piercing point}, where they orthogonally project onto coordinate planes. Surfaces intersect coordinate planes to make curves {trace curve}.
Four symmetry axes {quartic symmetry} can make eight angles of 45 degrees.
A line {transversal} can cross two coplanar lines, to form four angle pairs. Angles can orient in same direction {corresponding angle, transversal}. Angles {alternate angle} can lie on opposite sides of coplanar-line transversal. Angles can be opposite each other at intersection points {vertically opposite angle}.
If two transverse lines cross parallel lines, transversal line-segment ratios are equal {intercept theorem}.
Pappus invented line and centroid theorems {Pappus's theorems} {Pappus theorems}.
Draw line segments from first first-line-segment point to first second-line-segment point, second first-line-segment point to second second-line-segment point, and first first-line-segment point to second second-line-segment point. The drawn line segments have three intersection points that make a straight line.
Arcs have planes. Arcs rotated around plane axis generate area equal to arc length times circular-path segment traveled by arc centroid.
Planes rotated around plane axis generate volume equal to surface area times circular-path segment traveled by centroid.
revolution
If centroid paths are circles, the previous two theorems generate surfaces or volumes of revolution.
Two straight lines {anti-parallel line} can cut two straight lines so first angle cut by one line is supplementary to second angle cut by other line.
Line segments {extended line} can be longer, at either end.
Curved lines {isotopic line} can be perpendicular to themselves.
A line {median, trapezoid} joins midpoints of non-parallel trapezium or trapezoid sides.
Figure lines {nodal line} can stay fixed during rotation or deformation.
Lines {oblique line} can be neither parallel nor perpendicular to a direction.
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Date Modified: 2022.0225