User:TOstojich/College Geometry/Basics

Points and Lines
The uppercase Roman letters (A, B, C ...) will be used to denote points. In addition to denoting points in space, they can also denote the vertexes of a polygon and the vertexes' respective angles. For two arbitrary points A and B:


 * [$\overline{AB}$] or $\overline{AB}$ denotes a line segment beginning at A and ending at B. Line segments are geometric objects defined to be the shortest path between two points. In other words, if you had two points on a piece of paper, and you wanted to trace the shortest possible path between those two points, the result will always be a line segment. Here, the left square bracket and the right square bracket mean that the line segment begins at A and ends at B.


 * [$\overline{AB}$) denotes a ray that begins at A and passes through B. Rays are like a line segment, except that rays don't just stop when they reach B like line segments do; instead, they keep on going to infinity in one direction. Because of this, we use a left square bracket to enclose the A and a right parenthesis to enclose the B.


 * ($\overline{AB}$) denotes a line beginning at A and ending at B. Lines are like rays, except that rays go to infinity in both directions. Because of this, both A and B are surrounded by parenthesis instead of square brackets.

Most other geometry textbooks try to use $$\overrightarrow{AB}$$ and $$\overleftrightarrow{AB}$$ for rays and lines. This text chooses a different notation because it's more typographically convenient and easier to represent in HTML.

Polygons and Shapes
Polygons are general shapes where three or more line segments are connected together to form a closed loop. These line segments are called sides. When these sides are connected, they form vertexes. This book will denote a polygon by its vertexes, and it will denote them by a square character (□) following with the name of the vertexes. So if a polygon has five vertexes named alphabetically, it shall be titled □ABCDE. Sides of a polygon shall be denoted by lowercase roman letters (a, b, c...).

A special case of polygons are known as the triangles, which are polygons with three sides. The sum of the sides' lengths is commonly called the perimeter. In this text, the perimeter will be denoted by 2p; the constant "2" appears so that we can denote the semiperimeter (that is, the perimeter divided by two) by simply p. Triangles shall be denoted by △ABC, where A, B, and C refer to the vertexes of the triangle.

ha, hb, and hc denote the altitudes and ma, mb, and mc the medians of △ABC, with the subscripts referring to sides a, b, and c respectively. Altitudes are line segments perpendicular to a respective side which connect to the opposite vertex. Medians are line segments that connect its respective midpoint to the opposite vertex of a triangle.

ta is the internal bisector of ∠A. Ta is the external bisector of ∠A. The difference between internal and external bisectors can be found in the figure to the right.

Circles
Circles are geometric objects with a very specific definition: they are the set of points (often called a locus) that have the same distance from a given point, with the name of the point being the center of the circle. Circles with a specific center and radius will be denoted by ⨀$\overline{OR}$, where $\overline{OR}$ is the radius and the terminal point O is the center. If the value of the radius is dependent on a line defined elsewhere, then a circle can be denoted by ⨀(O, $\overline{AB}$), where O is still the center and $\overline{AB}$ is the line being used for the radius.