Clarification of geometrical notations: $|AB| $ vs $\overline{AB}$.

Question

1. So assume there is a line from point $A$ to point $B$. The line is, in some papers, denoted to be $\overline{AB}$ and the length of the line is denoted $|AB|$. But should not the length of the line be $|\overline{AB}|$?

2. Also, what if the points $A,B$ are placed on the periphery of a circle and I want to denote the arc and the arc length of the line segment between $A$ and $B$, what is the mathematical notation for these?

3. Assume there is a triangle with corners at $A,B,C$ and I want to denote the line from $A$ to $C$, is it $AC$ or $CA$? Is there any convention that the first letter is the letter that comes first in the alphabet or are the two equivalent and interchangeable? Same question for denoting the triangle, is it $ABC$, $BCA$, $CBA$ or $CAB$?

Answer 1

Good questions. I am not sure there are universal answers, so these reflect what I'd typically use.

1. You probably mean a line segment joining $A$ and $B$ vs. a line through them which could have any length including infinite. For this $\overline{AB}$ is fine, and for the length (measure) I learned it as $m\overline{AB}$ vs. $|\overline{AB}|$ although the latter seems reasonable by analogy with absolute value. Agree not $|AB|$, but most would interpret as the length anyway.

2. This seems pretty clear: $\overset\frown{AB}$ and $m\overset\frown{AB}$.

(\overarc known to be buggy in MathJax so this is close as I could get)

3. I think it's more common for $\overline{AB}$ to mean $A$ to $B$ where direction actually matters (it may not in many high-school geometry contexts but it's still good to have a convention). For the triangle it does not matter except when comparing to another as in a congruence proof then you'd like to express as $ABC$ and $DEF$ in a consistent way - meaning once defined you then compare "like" sides correctly. And don't forget that for $\angle ABC$ $B$ is the vertex (always).

Hope this is helpful.