# Beginning Terminology: Points, Lines, and Planes

The word ‘geometry’ derives from the Greek words for ‘earth’ (geo) and ‘to measure’ (metron).

A *construction* is a
geometric drawing for which only a compass
and a straightedge may be used.
You will study many different
constructions in Geometry, many of which are
beautifully illustrated here:

Zef Damen Constructions with Ruler and Compass

a compass:

for making circles;
for transferring the distance
between two existing points

ruler/straightedge:

used for drawing a line between two existing points;
no distance markings will be used;
no measurements are allowed

A *conjecture* is an educated guess.

Using specific observations
and examples to arrive at a conjecture
is called *inductive reasoning*.
For example, you might make the conjecture
that for all real numbers $\,x\,$ and
$\,y\,,$ the distance between them is
given by the formula $\,y-x\,.$

A *counterexample* is a
specific example that shows that a conjecture
is not always true.
For example, here is a counterexample
to the previous conjecture:

Let $\,y=5\,$ and $\,x=7\,.$ Then, $\,y-x=5-7=-2\,,$ which is not the distance between them.

*Deductive reasoning* uses logic,
and statements that are already accepted to be true,
to reach conclusions.
The methods of mathematical proof
are based on deductive reasoning.

*Point*, *line*, and *plane*
are three *undefined terms*
to get us started in the study of geometry—we will just agree on
their meaning.

A *point* represents
an exact location.
It is represented with a dot.
Capital letters, like $\,P\,,$
are frequently used to denote points.

*Space* is the set of *all* points.

A *geometric figure* is a subset of space.

That is, a geometric figure is
*any* collection of points.
Of course, there are certain important
geometric figures (like triangles and circles)
that will be studied throughout the course.

A *line* has length only;
it has no width or thickness;
it extends forever in both directions.
A line will be denoted using a lowercase
script letter, like $\,\ell \,.$

If $\,A\,$ and $\,B\,$ are two distinct points, then they determine a unique line which will be denoted by $\,\overleftrightarrow{AB}\,$ or $\,\overleftrightarrow{BA}\,.$

A *plane* is a flat surface
that extends infinitely in all directions;
it has length and width only;
it has no thickness.
A plane will be denoted using an
uppercase script letter, like $\,\mathcal{P}\,.$

If $\,A\,,$ $\,B\,,$ and $\,C\,$ are three distinct noncollinear points (see below), then they determine a unique plane which will be denoted by $\,ABC\,.$

Note: In the following definitions, the prefix ‘co’ means ‘same’.

*Collinear points*:
three or more points lying
on the same line

*Noncollinear points*:
points not lying on the same line

*Coplanar points*:
points lying in the same plane

*Noncoplanar points*:
points not lying in the same plane

*Coplanar* can also refer
to other geometric figures.
For example,
two lines are coplanar
if and only if
they lie in the same plane.

Three or more lines are *concurrent* if they share a unique common point.