# 1.1: Finite Geometries

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## Definition: Intersect

Two lines intersect if and only if they share a point.

## Definition: Parallel

Two lines are parallel if and only if they do not intersect.

## Definition: Four Point Geometry

The four point geometry is defined by the following axioms and definitions.

1. There exist exactly four points.
2. Any two distinct points have exactly one line on both of them.
3. Each line is on exactly two points.

Explore the four point geometry as follows.

1. Draw and label four points.
2. Use axiom 2 to draw as many lines as possible.
3. How many lines exist in this geometry?
4. Find a pair of parallel lines.
5. Can you find three lines that are pairwise parallel?
6. Can you find a point that is on three lines?

This page titled 1.1: Finite Geometries is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Mark A. Fitch via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.