Points, Lines, Planes

IntroductionLink to introduction

Geometry begins not with shapes or formulas, but with three primitive ideas so fundamental they cannot be defined in terms of anything simpler: the point, the line, and the plane. These are the atoms of all geometric reasoning — everything from triangles to spheres is ultimately built from them.

In this lesson we lay the rigorous groundwork of Euclidean geometry, exploring how these undefined objects behave, how they relate to one another, and how the postulates that govern them give rise to all the theorems you will encounter throughout this course.

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1. Undefined TermsLink to 1-undefined-terms

1.1 PointLink to 1-undefined-terms-1-1-point

A point has no dimension — no length, no width, no depth. It represents a precise location in space.

• Notation: capital letters $A$, $B$, $P$, $Q$
• A point has zero measure in every direction: its "size" is $0$

Although a point has no size, we draw it as a small dot as a visual aid. Formally:

$\dim(\text{point}) = 0$

1.2 LineLink to 1-undefined-terms-1-2-line

A line is a one-dimensional object extending infinitely in both directions. It has length but no width or thickness.

• Notation: lowercase letters $\ell$, $m$, $n$, or by two points on it: $\overleftrightarrow{AB}$
• A line contains infinitely many points
• Through any two distinct points there is exactly one line

$\dim(\text{line}) = 1$

A line segment $\overline{AB}$ is the finite portion of a line between (and including) two points $A$ and $B$. Its length is denoted $AB$ or $|\overline{AB}|$.

A ray $\overrightarrow{AB}$ starts at endpoint $A$ and extends infinitely through $B$.

1.3 PlaneLink to 1-undefined-terms-1-3-plane

A plane is a two-dimensional flat surface extending infinitely in all directions within that surface.

• Notation: capital script letters $\mathcal{P}$, $\mathcal{Q}$, or named by three non-collinear points: plane $ABC$
• A plane contains infinitely many lines and infinitely many points
• Through any three non-collinear points there is exactly one plane

$\dim(\text{plane}) = 2$

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2. Euclid's PostulatesLink to 2-euclids-postulates

Euclid's five postulates form the logical foundation of classical geometry. The first three are directly about our three primitive objects.

These postulates are not proved — they are assumed. Every theorem in Euclidean geometry follows logically from them.

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3. Collinearity and CoplanarityLink to 3-collinearity-and-coplanarity

3.1 Collinear PointsLink to 3-collinearity-and-coplanarity-3-1-collinear-points

Points are collinear if they all lie on the same line.

Test for collinearity (coordinate form): Points $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$ are collinear if and only if:

$\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0$

Expanding this determinant:

$\begin{aligned} &x_1(y_2 - y_3) - y_1(x_2 - x_3) + (x_2 y_3 - x_3 y_2) = 0 \end{aligned}$

Example: Are $A(1, 2)$, $B(3, 6)$, $C(5, 10)$ collinear?

$\begin{aligned} &1(6 - 10) - 2(3 - 5) + (3 \cdot 10 - 5 \cdot 6) \\ &= 1(-4) - 2(-2) + (30 - 30) \\ &= -4 + 4 + 0 \\ &= \boxed{0} \end{aligned}$

Yes — the three points are collinear. They all lie on the line $y = 2x$.

3.2 Coplanar PointsLink to 3-collinearity-and-coplanarity-3-2-coplanar-points

Points (or lines) are coplanar if they all lie in the same plane.

• Any two points are always coplanar (infinitely many planes contain any line)
• Any three points are always coplanar (exactly one plane contains three non-collinear points)
• Four or more points may or may not be coplanar

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4. IntersectionsLink to 4-intersections

How do our three primitive objects intersect each other? The answer depends on their relative positions.

4.1 Two LinesLink to 4-intersections-4-1-two-lines

Two distinct lines in the same plane either:

In three-dimensional space, a third possibility exists:

| Skew | Non-parallel, non-intersecting | $\emptyset$ |

Two lines $\ell_1$ and $\ell_2$ are parallel, written $\ell_1 \parallel \ell_2$, if they lie in the same plane and do not intersect.

4.2 Line and PlaneLink to 4-intersections-4-2-line-and-plane

A line and a plane in space either:

A line is perpendicular to a plane if it is perpendicular to every line in the plane that passes through the point of intersection. Written $\ell \perp \mathcal{P}$.

4.3 Two PlanesLink to 4-intersections-4-3-two-planes

Two distinct planes either:

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5. Distance and the Ruler PostulateLink to 5-distance-and-the-ruler-postulate

5.1 The Ruler PostulateLink to 5-distance-and-the-ruler-postulate-5-1-the-ruler-postulate

The Ruler Postulate states: the points on any line can be placed in a one-to-one correspondence with the real numbers $\mathbb{R}$, such that the distance between any two points $A$ and $B$ equals the absolute value of the difference of their coordinates.

If $A$ has coordinate $a$ and $B$ has coordinate $b$, then:

$AB = |a - b|$

This single postulate guarantees that distance is always non-negative, symmetric ($AB = BA$), and zero only when $A = B$:

$\begin{aligned} AB &\geq 0 \\ AB &= BA \\ AB &= 0 \iff A = B \end{aligned}$

5.2 The Segment Addition PostulateLink to 5-distance-and-the-ruler-postulate-5-2-the-segment-addition-postulate

If point $B$ lies between $A$ and $C$ on a line, then:

$\boxed{AB + BC = AC}$

This seemingly obvious statement is crucial: it lets us decompose and combine lengths algebraically.

Example: Given $AC = 14$ and $AB = 2x - 1$, $BC = x + 4$. Find $x$ and all segment lengths.

$\begin{aligned} AB + BC &= AC \\ (2x - 1) + (x + 4) &= 14 \\ 3x + 3 &= 14 \\ 3x &= 11 \\ x &= \frac{11}{3} \end{aligned}$

$AB = 2\left(\frac{11}{3}\right) - 1 = \frac{22}{3} - \frac{3}{3} = \frac{19}{3}, \qquad BC = \frac{11}{3} + 4 = \frac{23}{3}$

Verification: $\dfrac{19}{3} + \dfrac{23}{3} = \dfrac{42}{3} = 14$ ✓

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6. Midpoints and BisectorsLink to 6-midpoints-and-bisectors

6.1 Midpoint DefinitionLink to 6-midpoints-and-bisectors-6-1-midpoint-definition

The midpoint $M$ of segment $\overline{AB}$ is the unique point on $\overline{AB}$ such that:

$AM = MB = \frac{AB}{2}$

In coordinate geometry, if $A = (x_1, y_1)$ and $B = (x_2, y_2)$, then:

$M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$

More generally in $\mathbb{R}^n$, the midpoint of $\mathbf{a}$ and $\mathbf{b}$ is:

$M = \frac{\mathbf{a} + \mathbf{b}}{2}$

6.2 Segment BisectorLink to 6-midpoints-and-bisectors-6-2-segment-bisector

A segment bisector is any line, ray, segment, or plane that passes through the midpoint of a segment. It divides the segment into two congruent halves.

The perpendicular bisector of $\overline{AB}$ is the unique line that:

• Passes through the midpoint $M$ of $\overline{AB}$
• Is perpendicular to $\overline{AB}$

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7. Angles — Where Lines MeetLink to 7-angles-where-lines-meet

When two rays share a common endpoint, they form an angle.

7.1 DefinitionLink to 7-angles-where-lines-meet-7-1-definition

An angle $\angle BAC$ (or $\angle A$) consists of:

• Two rays (called the sides): $\overrightarrow{AB}$ and $\overrightarrow{AC}$
• A common endpoint called the vertex: $A$

The measure of an angle, written $m\angle BAC$, is a real number in $[0°, 360°]$ (or $[0, 2\pi]$ radians).

The conversion between degrees and radians is:

$\theta_{\text{rad}} = \theta_{\deg} \cdot \frac{\pi}{180°}, \qquad \theta_{\deg} = \theta_{\text{rad}} \cdot \frac{180°}{\pi}$

7.2 Angle ClassificationLink to 7-angles-where-lines-meet-7-2-angle-classification

7.3 The Angle Addition PostulateLink to 7-angles-where-lines-meet-7-3-the-angle-addition-postulate

If ray $\overrightarrow{BD}$ lies in the interior of $\angle ABC$, then:

$\boxed{m\angle ABD + m\angle DBC = m\angle ABC}$

This mirrors the Segment Addition Postulate precisely — angles can be decomposed and combined just like lengths.

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8. Special Angle PairsLink to 8-special-angle-pairs

Two angles may have a special relationship based on their measures or positions.

Theorem — Vertical Angles are Congruent:

Proof. Let lines $\ell_1$ and $\ell_2$ intersect at point $P$, forming angles $\angle 1$, $\angle 2$, $\angle 3$, $\angle 4$ in order around $P$. Then $\angle 1$ and $\angle 2$ form a linear pair, so:

$m\angle 1 + m\angle 2 = 180°$

Similarly, $\angle 2$ and $\angle 3$ form a linear pair:

$m\angle 2 + m\angle 3 = 180°$

Subtracting the first equation from the second:

$m\angle 3 - m\angle 1 = 0 \implies \boxed{m\angle 1 = m\angle 3}$

By identical reasoning, $m\angle 2 = m\angle 4$. $\blacksquare$

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9. Points and Lines in Three-Dimensional SpaceLink to 9-points-and-lines-in-three-dimensional-space

Everything so far has lived in a single plane. In three dimensions, the relationships become richer.

9.1 Determining a PlaneLink to 9-points-and-lines-in-three-dimensional-space-9-1-determining-a-plane

A unique plane is determined by each of the following:

1. Three non-collinear points $A$, $B$, $C$
2. A line and a point not on the line: $\ell$ and $P \notin \ell$
3. Two intersecting lines: $\ell_1 \cap \ell_2 = \{P\}$
4. Two parallel lines: $\ell_1 \parallel \ell_2$

9.2 Skew LinesLink to 9-points-and-lines-in-three-dimensional-space-9-2-skew-lines

Lines $\ell_1$ and $\ell_2$ are skew if:

• They are not parallel (different directions), and
• They do not intersect

This is only possible in three or more dimensions. The distance between skew lines is the length of the unique segment perpendicular to both.

Formula: Given skew lines with direction vectors $\mathbf{d}_1$ and $\mathbf{d}_2$ passing through points $\mathbf{p}_1$ and $\mathbf{p}_2$:

$d = \frac{|(\mathbf{p}_2 - \mathbf{p}_1) \cdot (\mathbf{d}_1 \times \mathbf{d}_2)|}{|\mathbf{d}_1 \times \mathbf{d}_2|}$

where $\times$ denotes the cross product and $\cdot$ the dot product. We will derive this formula fully in Chapter 14 on vectors.

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10. The Coordinate Model — Connecting Algebra and GeometryLink to 10-the-coordinate-model-connecting-algebra-and-geometry

René Descartes unified geometry and algebra through the Cartesian coordinate system, placing points in correspondence with ordered pairs (or triples) of real numbers.

In $\mathbb{R}^2$: a point is $(x, y) \in \mathbb{R}^2$

In $\mathbb{R}^3$: a point is $(x, y, z) \in \mathbb{R}^3$

The distance formula in $\mathbb{R}^2$ follows directly from the Pythagorean theorem:

$d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

In $\mathbb{R}^3$:

$d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

These formulas are both instances of the general Euclidean norm in $\mathbb{R}^n$:

$d(\mathbf{a}, \mathbf{b}) = \|\mathbf{a} - \mathbf{b}\| = \sqrt{\sum_{i=1}^{n}(a_i - b_i)^2}$

Proof of the 3D distance formula:

Let $A = (x_1, y_1, z_1)$ and $B = (x_2, y_2, z_2)$. Let $C = (x_2, y_2, z_1)$ — the point directly below $B$ at the same height as $A$.

Step 1: Distance $AC$ lies in the horizontal plane: $AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 2: $\overline{AB}$ is the hypotenuse of right triangle $ACB$ with legs $AC$ and $CB = |z_2 - z_1|$: $\begin{aligned} AB^2 &= AC^2 + CB^2 \\ &= (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2 \end{aligned}$

$\boxed{AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}} \qquad \blacksquare$

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SummaryLink to summary

The three undefined terms — point, line, and plane — are the irreducible building blocks of Euclidean geometry. From them, and from Euclid's five postulates, we have established:

• Collinearity and coplanarity as properties of point sets
• The Ruler Postulate grounding distance in the real numbers
• The Segment Addition and Angle Addition postulates for decomposing lengths and angles
• Midpoints, bisectors, and special angle pairs as fundamental constructions
• Skew lines as a genuinely three-dimensional phenomenon
• The distance formula as an algebraic encoding of the Pythagorean theorem

In the next lesson, Angle Relationships, we will build directly on the special angle pairs introduced here and develop the full theory of angles formed by parallel lines cut by a transversal.