Creating Euclidean Geometry

Last time, we discussed symmetric bilinear forms and its applications into geometry, in particular, to rational trigonometry, which posits a computationally tractable alternative to classical trigonometry. The rational analog of distance $d$ would be the quadrance $Q$ , and the rational analog of angle $\theta$ would be the spread $s$ , connected in the case $V = \mathbb R^2$ by the equations

$Q = d^2,\quad s = \sin^2\theta.$

In this post, we see how we can use linear algebra and ideas in rational trigonometry to define a broad framework for geometry, of which we obtain Euclid’s five postulates in the special case $V = \mathbb R^2$ .

Let $V$ be a vector space over a field $\mathbb K$ .

Definition 1. An affine space $\mathbb A$ over $V$ is a set $\mathbb A$ , whose elements are called points, equipped with an addition map $+ : \mathbb A \times V \to \mathbb A$ that satisfies the following properties:

For any $A \in \mathbb A$ , $A + \mathbf 0 = A$ .
For any $A \in \mathbb A$ and $\mathbf v , \mathbf w \in V$ , $(A + \mathbf v) + \mathbf w = A + (\mathbf v + \mathbf w)$ .
For any $A \in \mathbb A$ , the map $V \to \mathbb A, \mathbf v \mapsto A + \mathbf v$ is bijective.

Example 1. Any vector space $V$ is an affine space over itself. Defining $\mathbb A = V$ , for each point $A$ in $\mathbb A$ , we will define the position vector $\overrightarrow{OA} = A \in V$ .

Lemma 1. For points $A, B \in \mathbb A$ , there exists a unique vector $\overrightarrow{AB} \in V$ such that $A + \overrightarrow{AB} = B$ .

Proof. Fix $A, B \in \mathbb A$ . Since $\mathbf v \mapsto A + \mathbf v$ is bijective, there exists a unique vector $\mathbf v \in V$ such that $A + \mathbf v = B$ . Denote $\mathbf v = \overrightarrow{AB}$ .

Hence, we will denote $\overrightarrow{AB} = B - A = \overrightarrow{OB} - \overrightarrow{OA}$ without loss of ambiguity.

Lemma 2. For points $A, B, C, D \in \mathbb A$ , $\overrightarrow{AB} = \overrightarrow{CD}$ precisely when $A + \overrightarrow{CD} = B$ and $C + \overrightarrow{AB} = D$ .

Henceforth, let $\mathbb A$ denote any affine space over $V$ . Therefore, points are positions determined by vectors that start from the origin. Each point $R \in \mathbb A$ can be described by a position vector $\mathbf r = \overrightarrow{OR}$ . Let $\mathbb K$ be any field that contains $\mathbb Q$ .

Definition 1. A line $l$ containing the point $A \in \mathbb A$ and parallel to the vector $\mathbf d \in V$ is the set

$l := \{A + t \mathbf d : t \in \mathbb K\} \subseteq \mathbb A.$

Equivalently, we say that $R$ belongs in the line if its position vector $\mathbf r = \mathbf r(t)$ satisfies the parametric equation of a line, i.e. $l : \mathbf r(t) = \mathbf a + t \mathbf d$ .

Theorem 1. For any two distinct points $A, B \in \mathbb A$ , there exists a unique line that contains $A$ and $B$ .

Proof. For existence, define the line $l_{AB}$ by the equation $\mathbf r(t) = \mathbf a + t(\mathbf b - \mathbf a)$ . Since $\mathbf r(0) = \mathbf a$ and $\mathbf r(1) = \mathbf b$ , $A,B$ lie on $l_{AB}$ .

For uniqueness, let $m$ be any other line that passes through $A, B$ . Then there exists a point $C \in \mathbb A$ and a vector $\mathbf u \in V$ such that $m : \mathbf r(s) = \mathbf c + s\mathbf u$ . Find distinct scalars $s_0, s_1 \in \mathbb K$ such that $\mathbf c + s_0 \mathbf u = \mathbf a$ and $\mathbf c + s_1 \mathbf u = \mathbf b$ . Then

$\displaystyle \mathbf b - \mathbf a = (s_1 - s_0)\mathbf u \quad \Rightarrow \quad \mathbf u = \frac 1{s_1-s_0} (\mathbf b - \mathbf a).$

We need to prove that $m = l_{AB}$ . For any $F \in m$ , find $s_3 \in \mathbb K$ such that

$\displaystyle \mathbf f = \mathbf c + s_3 \mathbf u = \mathbf a + (s_3 - s_0)\mathbf u = \mathbf a + \frac{s_3 - s_0}{s_1 - s_0} (\mathbf b - \mathbf a)\quad \Rightarrow \quad F \in l_{AB}.$

Hence, $m \subseteq l_{AB}$ . We can prove $l_{AB} \subseteq m$ similarly since

$\mathbf a + t(\mathbf b - \mathbf a) = \mathbf c + (s_0 + ts_1 - ts_0)\mathbf u.$

In particular, when $\mathbf A = V = \mathbb R^2$ and $\mathbb K = \mathbb R$ , we obtain Euclid’s first two postulates:

Corollary 1. A straight line segment can be drawn joining any two points. Furthermore, any straight line segment can be extended indefinitely in a straight line.

The rest of Euclid’s postulates will require our previous discussion on symmetric bilinear forms $[ \cdot , \cdot ]$ , which basically characterise perpendicularity, since the usual inner product $\langle \cdot, \cdot \rangle$ on $\mathbb R^n$ yields the desired properties.

Recall that for any symmetric bilinear form $[ \cdot , \cdot ] : V \times V \to \mathbb K$ , we can define the quadrance $Q : V \to \mathbb K$ by $Q(\mathbf v) = [ \mathbf v , \mathbf v ]$ . In particular, for points $A , B \in \mathbb A$ , we can define the quadrance between two points $Q : \mathbb A \times \mathbb A \to \mathbb K$ by $Q(A,B) := Q(\overrightarrow{AB})$ .

Definition 3. A sphere $S \subseteq \mathbb A$ with centre $C$ and quadrance $Q$ is defined to be the set

$S := \{R \in \mathbb A : Q(C,R) = Q\} \subseteq \mathbb A.$

A sphere in $\mathbb A = V = \mathbb R^2$ equipped with the usual inner product is called a circle.

Theorem 2. For any two distinct points $A, B \in \mathbb A$ , there exists a unique circle with centre $A$ and quadrance $Q(A,B)$ .

Corollary 2. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

Definition 4. For $\mathbf u, \mathbf v \in V$ , we say that $\mathbf u$ and $\mathbf v$ are perpendicular or orthogonal, denoted $\mathbf u \perp \mathbf v$ , if $[\mathbf u, \mathbf v] = 0$ .

Lemma 3. If $\mathbf u, \mathbf v$ have nonzero quadrances, then $\mathbf u \perp \mathbf v \iff s(\mathbf u, \mathbf v) = 1$ .

Theorem 3. All right angles are congruent to one another.

We are left with the infamous parallel postulate, which seems nontrivial to state, but turns out to be definable using the notions we discussed in affine spaces. We remark that to debunk the parallel postulate, we will need a broader notion of space, possibly projective space, to account for the varied notions of parallelism.

Definition 5. Two nonzero vectors $\mathbf u, \mathbf v$ are parallel, denoted $\mathbf u \parallel \mathbf v$ , if the set $\{\mathbf u, \mathbf v\}$ is linearly dependent i.e. there exists some $\alpha \in\mathbb K$ such that $\mathbf v = \alpha \mathbf u$ . We say that the lines $l_{AB}$ and $l_{CD}$ are parallel if $\overrightarrow{AB} \parallel \overrightarrow{CD}$ .

Theorem 4. For any line $l$ and point $A$ , there exists at most one line passing through $A$ that is parallel to $l$ .

Proof. Let $l_1$ and $l_2$ be lines that pass through $A$ and are parallel to $l$ . Let $B,C$ be points on $l$ and define $\mathbf d := \overrightarrow{BC}$ . For any point $R$ on $l_1$ , $\overrightarrow{AR} \parallel \mathbf d$ . Likewise, for any point $S$ on $l_2$ , $\overrightarrow{AS} \parallel \mathbf d$ . Thus, $\overrightarrow{AR} \parallel \overrightarrow{AS}$ . Some bookkeeping yields $l_1 = l_2$ .

Corollary 3 (Playfair’s Axiom). In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

Corollary 4. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Proof. Corollary 3 implies Corollary 4 via the following geometric argument, which only requires Corollary 1 and Corollary 2 to establish.

With that, we have successfully created a working model for Euclidean geometry, defining quadrances and spreads in general fields, then take square roots in the case $\mathbb K = \mathbb R$ and even employing trigonometric functions to obtain our classical notions of distances and angles (in radians). We now list three common starting points for elementary school students’ learning of Euclidean geometry.

Corollary 5. Define $1^{\circ} := \pi/180$ so that $180^{\circ} = \pi$ . The following proposition holds:

Adjacent angles on a straight line sum to $180^{\circ}$ .
Interior angles on the same side of a transversal sum to $180^{\circ}$ .

Interestingly, we do not have purely mechanistic tools to draw an angle of $1^{\circ}$ , but can define the concept mathematically and obtain close approximations to that. However, using constructions in Euclidean geometry, we can draw the obvious angle ${180}^\circ$ and the special angle $60^{\circ}$ , bisections of these angles, sums and differences, and integer multiples of said angles.

In the next post, we take advantage of inner product spaces to discuss orthogonal functions, and derive the usual formulas involved in discussing Fourier series.

—Joel Kindiak, 20 Mar 25, 1523H

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Baby Linear Algebra – KindiakMath

December 17, 2025 at 9:21 pm

[…] will talk about some basic ideas about lines and angles, which can be described using lines, and be precisely formulated using—you guessed it—linear algebra. 90% of my blog was dedicated to ensure that what we do at […]

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Creating Euclidean Geometry

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