Points, Lines & Planes

In geometry we start with three ideas so basic that we do not even define them. We describe them instead.

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• Point --- a single exact location. It has no size. We draw a dot and name it with a capital letter, like $P$.
• Line --- a straight path of points that goes on forever in both directions. Name it by any two of its points with a line symbol, $\overleftrightarrow{AB}$, or by a lowercase letter, $\ell$.
• Plane --- a perfectly flat surface that goes on forever. Name it with a capital script letter or by three points that are not all on one line, like plane $ABC$.

Collinear points lie on the same line. Coplanar points lie on the same plane.

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• A segment $\overline{AB}$ is the part of a line between two endpoints $A$ and $B$, including both endpoints. It has a definite length.
• A ray $\overrightarrow{AB}$ starts at endpoint $A$ and goes forever through $B$. Order matters: $\overrightarrow{AB}$ and $\overrightarrow{BA}$ point opposite ways.
• Opposite rays are two rays with the same endpoint that together form a straight line.

Notation: $\overline{AB}$ names the segment (a figure); $AB$ with no bar means its length (a number).

An angle measures a turn.

If point $B$ lies on $\overline{AC}$ (between $A$ and $C$), then the two shorter lengths add to the whole:

This lets you find any one of the three lengths when you know the other two.

Two segments are congruent ($\overline{AB} \cong \overline{CD}$) when they have equal length ($AB = CD$). We mark congruent segments with matching tick marks.

The midpoint $M$ of $\overline{AB}$ is the point exactly halfway between $A$ and $B$. It splits the segment into two congruent halves:

For points $A(x_1,y_1)$ and $B(x_2,y_2)$:

The midpoint averages the coordinates. The distance formula is the Pythagorean Theorem in disguise.

A segment bisector is any line, ray, or segment that passes through the midpoint of a segment, cutting it into two congruent halves. A line that bisects a segment and meets it at a right angle is a perpendicular bisector.

Copying a segment $\overline{AB}$: Draw a ray with endpoint $C$. Open your compass to the length $AB$. Without changing that opening, place the point on $C$ and swing a small arc across the ray; label the crossing $D$. Then $\overline{CD} \cong \overline{AB}$.

Bisecting a segment $\overline{AB}$: Open the compass wider than half of $AB$. From $A$, draw arcs above and below the segment; from $B$ with the same opening, draw two more arcs. Connect the two crossing points. That line is the perpendicular bisector, and where it meets $\overline{AB}$ is the midpoint.

The midpoint splits a segment in the ratio $1:1$. To find a point $P$ that divides $\overline{AB}$ in any ratio $m:n$ (so that $AP:PB = m:n$), with $A(x_1,y_1)$ and $B(x_2,y_2)$, use the section formula:

Notice the “cross” pattern: the weight $m$ (nearest to $B$) multiplies the $B$-coordinates, and $n$ multiplies the $A$-coordinates. Setting $m=n=1$ recovers the midpoint formula, since each coordinate becomes a plain average.

A point in space needs three coordinates, $A(x_1,y_1,z_1)$. The plane formulas simply grow a $z$-term:

This is still just the Pythagorean Theorem, applied twice: once in the base plane and once up the vertical direction.

When points and lines are in “general position” (no needless overlaps), the counts follow neat patterns. For $n$ points with no three collinear, the number of distinct lines through pairs of them is

For $n$ points with no four coplanar, the number of distinct planes through triples of them is

And $n$ lines drawn in a plane, no two parallel and no three meeting at one point, cut the plane into

regions --- each new line adds one more region than the line before it.

To find the shortest path from point $A$ to a line $\ell$ and then on to point $B$ (both $A$ and $B$ on the same side of $\ell$), reflect one point across the line. If $A'$ is the mirror image of $A$ across $\ell$, then for any point $X$ on the line $AX = A'X$. So the bent path $A \to X \to B$ has the same length as $A' \to X \to B$, which is shortest when $A'$, $X$, and $B$ are collinear. Draw the straight segment $\overline{A'B}$; where it crosses $\ell$ is the best point $X$.

A locus is the set of all points that satisfy a given condition, and nothing else. Two rules of thumb turn everyday shapes into loci:

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• The locus of points a fixed distance $r$ from a point $O$ is a circle of radius $r$ centered at $O$.
• The locus of points equidistant from two fixed points $A$ and $B$ is the perpendicular bisector of $\overline{AB}$.

To describe a locus, ask: what shape do all the qualifying points trace out?