# Easy, early proofs in Young's geometry

The axioms for Young's geometry (finite affine plane of order 3) appear as follows in the text:
1. There exists at least one line.
2. There are exactly three points on every line.
3. Not all points are on the same line.
4. There is exactly one line on any two distinct points.
5. For each line l and each point P not on l, there exists exactly one line on P which is not on any point of l.

Proposition 1. For every point, there is a line not on that point.

Proposition 2. For every point, there are at least four lines on that point.

Proposition 3. For every point, there are exactly four lines on that point.

Proposition 4. In Young's geometry, each line is parallel to exactly two lines.

Proposition 5. In Young's geometry, there are exactly 12 lines.

Proposition 6. In Young's geometry, there are three lines, no two of which intersect.

Proposition 7.In Young's geometry, there are exactly 9 points.

Proposition 8. Given any three mutually parallel lines in a Young's geometry, every other line intersects all three.

Len Smiley
Last modified: Tue Jan 23 11:02:08 AKST 2001