1. Write the plane dual of the axioms for the three-point geometry of Section 1.3

Solution:

1. There exist exactly 3 lines in this geometry.
2. Two distinct lines have exactly one point in common.
3. Not all the lines of the geometry pass through the same point.

4.Which axioms are also true statements in Euclidean geometry?

Ans: None. Solution: There are an infinite number of lines in Euclidean geometry. There exist parallel lines in Euclidean geometry. Each point is on an infinite number of lines in Euclidean geometry.

6. Do each two points of the geometry lie on a common line?.

Ans: No. Solution: In this geometry there are 6 points and every line has 3 points on it. The number of pairs of points is C(6,2) = 15 and each line contributes 3 pairs to this total. Thus, it would take 5 lines for every pair of points to lie on a common line, and the geometry only has 4 lines. In Fig. 1.6 (pg. 13), the points C and D do not lie on a common line.

9. How many other lines are parallel to each line?

Ans: None. Solution:By Axiom 2, any pair of distinct lines meet, and so, are not parallel.

11. Prove that a set of two lines cannot contain all the points of the geometry.

Solution:Every line has three points on it. There are six points in the geometry. Given two lines that are distinct, they have a common point by Axiom 2. Thus, there are only 5 distinct points on any two lines, so no two lines can contain all the points of the geometry.

Exercises 13-24 refer to the four-point geometry.

13. Draw another model for this geometry different from those shown in Figure 1.7.

Solution:

15. Rewrite the set of axioms for this geometry, using tree for point and row for line.

Solution:

1. The number of trees in this geometry is four.
2. Any two distinct trees are contained in exactly one row.
3. Each row contains exactly two trees.

Solution: Since a line has only two points on it (Axiom 3), once two points are given, the other two points must form a line parallel to the one determined by the first two points (Axiom 2). So, the parallel lines are: AB and CD, AC and BD, and AD and BC.

21. Prove, without using the idea of duality, that the geometry includes exactly 6 lines.

23. Prove that a set of two lines exists that contains all the points of the geometry.

Solution: Given any line, it contains two points (Axiom 3), call them A and B. The other two points (since there are 4 all together - Axiom 1) will be called C and D. There is a line containing C and D (Axiom 2). This line and the first line contain all the points of the geometry.