- There exists at least one line.
- There are exactly three points on every line.
- Not all points are on the same line.
- There is exactly one line on any two distinct points.
- 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.

**Proof.** Let p be a point. We will find a line *l* which is not on p. By **Y1**, a line *l'* exists. If p is not on *l'*, we are done: take *l*=*l'*. If p is on *l'*, then by **Y2** there are two other points, p_{1} and p_{2}, on *l'*. By **Y3**, there is a point q not on *l'*. By **Y4**, there is a line *l''* on q and p_{1}. The point p is not on *l''* because if p and p_{1} were both on *l''*, then by **Y4**,
*l''* would be *l'*, but, by assumption, q is not on *l'*, while q is on *l''*. Take *l*=*l''*. QED

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

**Proof.** Let p be a point. Using Prop. 1, let *l* be a line which is not on p.
By Y2, there are 3 points, p_{1}, p_{2}, and p_{3}, on* l* . By Y4, for each pair {p,p_{1}}, {p,p_{2}}, and {p,p_{3}}, there is a
line* l*_{i} , i=1,2,3, on {p,p_{i} }. No two of these lines are the same, since, by Y4, if p_{i} and p_{j} are on a line, that
line must be* l* , but* l* is not on p. There is a fourth line,* l*_{4}, on p, by Y5. QED

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

**Proof.** Let p be a point. By the proof of Prop. 2, there is a line* l* not on p, lines* l*_{1},*l*_{2}, and* l*_{3} each on
p and each intersecting* l* , and a line* l*_{4} on p parallel to* l* . There can be no fifth line on p which is parallel to
l , since, by Y5,* l*_{4} is the only line with these properties. There can be no fifth line on p which intersects* l* ,
since, for instance, a line on p and p_{1} must be* l*_{1}, by Y4, and similarly for p_{2} and p_{3}. QED

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

**Proof.** Let* l* be a line. By Y3 it's on exactly 3 points, p_{1}, p_{2}, and p_{3}. By Prop. 2 there are 3 other
lines on p_{1}: call them* l*_{1},* l*_{2}, and* l*_{3}. By Y3, the line* l*_{1} is on two other (by Y4) points, q_{1} and q_{2}. By Y5, for
i=1,2, there is exactly one line m_{i} on q_{i} such that m_{i} is parallel to* l* . It's easy (but necessary) to show that
m_{1} and m_{2} are distinct. So there are at least two parallels to* l* . If there were another line m_{3} parallel to* l* ,
then by Y5,* l* would be the unique parallel to m_{3} on the point p_{1}. Then since* l*_{1} is on p_{1},* l*_{1} must intersect
m_{3}. The point of intersection is q_{1} or q_{2}. But then, by Y4, m_{3} would be either m_{1} or m_{2}, a contradiction.
QED

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

**Proof.** By Y1 there is a line* l* , which is on three points p_{1}, p_{2}, and p_{3}, by Y3. On each of these
points there are exactly three other lines by Prop. 3. No two of these additional lines can be the same, since
none may be on two points of* l* . So far we have 10 lines, and there can be no other line which intersects* l* .
By Prop. 4, there are exactly two lines parallel to* l* . Any 13th line must either intersect or be parallel to* l* ,
but this is impossible as noted. QED

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

**Proof.** We know there is a line* l* , and lines* l*_{1} and* l*_{2} parallel to* l* (by Y1 and Prop. 4). If* l*_{1}
intersected* l*_{2} at a point p, there would be two parallels to* l* through p, contradicting Y5. QED

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

**Proof.** Let* l*_{1} (on p_{1},p_{2},p_{3}),* l*_{2} (on p_{4},p_{5},p_{6})), and* l*_{3} (on p_{7},p_{8},p_{9})), be three lines as in Prop. 6. If
there were a 10th point, q, then by Y4 there is a line m on q and p_{1}. Since we already have a line through
p_{1} parallel to* l*_{2} , and a line through p_{1} parallel to* l*_{3} (in both cases, it's* l*_{1}), m must be on points of* l*_{2} and
*l*_{3}. This would force m to have at least four points, violating Y2. QED

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

**Proof.** Exercise.

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