Euclidean Plane is Abstract Geometry
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Theorem
The Euclidean plane $\struct {\R^2, L_E}$ is an abstract geometry.
Proof
We will show that the axioms for an abstract geometry hold.
Axiom 1
Let $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be two distinct points in $\struct {\R^2, L_E}$.
If $x_1 = x_2 = a$ then $P, Q \in L_a$.
If $x_1 \ne x_2$ then let:
- $m = \dfrac {y_2 - y_1} {x_2 - x_1}$
- $b = y_2 - m x_2$
Then $P, Q \in L_{m,b}$, since:
- $b + m x_1 = y_2 - m \paren {x_2 - x_1} = y_2 - \paren {y_2 - y_1} = y_1$
- $b + m x_2 = y_2 - m \paren {x_2 - x_2} = y_2$
So any two points in $\R^2$ lie on a line in $L_E$.
$\Box$
Axiom 2
Let $a \in \R$ be arbitrary.
Then:
- $\tuple {a, 0}, \tuple {a, 1} \in L_a$
Let $m, b \in \R$ be arbiitrary.
Then:
- $\tuple {0, b}, \tuple {1, m + b} \in L_{m, b}$
So every line in $L_E$ has at least two distinct points.
$\Box$
Hence $\struct {\R^2, L_E}$ is an abstract geometry.
$\blacksquare$
Sources
- 1991: Richard S. Millman and George D. Parker: Geometry: A Metric Approach with Models (2nd ed.) ... (previous) ... (next): $\S 2.1$