2. Which axioms in the geometry of Pappus are also true statements in Euclidean geometry?

Ans: Axioms 1, 3, 4, and 6 (with no exceptions).

3.For each point in the geometry of Pappus, how many other points in the geometry do not lie on a line through the given point?

Ans: 2. Solution: By Thm 1.10, there are 3 lines through each point. By axiom 2 there are 3 points on each line. So, a given point is joined to 6 other points by lines. Since there are 9 points in total, that leaves just 2 points that the original point is not joined to.

6. Prove that there are at least two lines in the geometry of Pappus parallel to a given line.

Solution:Let l be a given line. By Axiom 3, there is a point not on l, call it P. By Axiom 4, there is a line through P that is parallel to l. Since there are three points on l (Axiom 2), and P is not joined to only one of them (Axiom 5), there is a point on l that P is joined to. This line through P contains 3 points, let the other point that is not on l be called Q. By Axiom 4, again, there is a line through Q that is parallel to l. If this line were the same as the first parallel through P, it would contain both P and Q, and by Axiom 6 would have to be the line that intersects l, a contradiction since it is parallel to l. So, the two lines are distinct and both are parallel to l.

7. Prove that the geometry of Pappus contains exactly nine points.

Solution:There is a line, l, (Axiom 1), with three points on it (Axiom 2). Every point not on l is joined to two of these three points by lines (Axioms 5 and 6). Since there are three lines through a point (Thm 1.10), there are exactly 6 lines, other than l, that go through points of l. Each of these lines has two points not on l. This gives 6(2) = 12 possible points, but we have counted each point twice, so there are only 12/2 = 6 points not on l. Plus the 3 on l, gives 9 points in total.

The remaining exercises refer to the geometry of Desargues.

11. Prepare a table to represent the geometry of Desargues, using the points as named in Fig. 1.12 and letting each column of the table represent a line in the geometry.

Solution:

P	P	P	T	T	S	S	R	R	T
A	B	C	A	A'	A	A'	B	B'	R
A'	B'	C'	B	B'	C	C'	C	C'	S

16. In Fig. 1.12, name all the lines parallel to (line) AS.

Solution:There are 3 lines parallel to AS. PBB', TA'B' and RC'B'.

18. Prove that two lines parallel to the same line are not parallel to each other.

Solution: Let l be a fixed line with pole P (exists by Thm 1.11). Let m be any line parallel to l. If P is not on m, then by Axiom 6, m and l intersect. This contradicts the fact that l and m are parallel. So, any line parallel to l must pass through P. Therefore, two lines parallel to l must intersect at P, and so, can not be parallel to each other.

19. Prove there is a line through two distinct points if and only if their polars intersect.

Solution: Let P and Q be distinct points. Suppose that l is a line containing them both. Let R be the pole of l (exists by Thm 1.11). By exercise 17 (proved in class), since P is on the polar of R, R is on the polar of P. Similarly, R is on the polar of Q, so the polar of P and the polar of Q meet at the point R. Now, suppose only that the polar of P and the polar of Q meet at a point T. Since T is on the polar of P, P is on the polar of T. Similarly, Q is on the polar of T, and so, the polar of T contains both P and Q.