9+23


 * Proposition 2.10** If A*B*C and A*C*D then B*C*D and A*B*D.

Start by setting up the proposition with a picture. (The proofs shouldn't depend on drawings, nor should the drawings be an important part of the proof. We can refer to a drawing for clarification, but not as an essential part of the proof) Draw a line labeled P with the points **A*B*C*D** (can we say this?) on it. Then draw a line (not P) through the point C and label it L. Then draw another line M (not lines P or C) through point B.

By Lemma 2.8 A and D are on opposite sides of line L. By Lemma 2.6 A and B are on the same side of line L.

If B and D were on the same side of L then A and D would be on the same side by betweenness B-4

So B and D are on opposite sides because this isn't true By Lemma 2.9 B*C*D

same analysis with line m for A*B*D

Theorem 2.11** If C*A*B and L is the line through A,B, and C, then any point P lying on L lies either on the ray AB or on the ray AC.
 * Line Seperation Property

Start by drawing a picture. A line labeled L with three points on it C*A*B. Draw a line (not line L) through A and label it M. By lemma 2.8 C and B are opposite sides of M.

There are two options of how to pick point P: (Which ones?)

(1) By Lemma 2.9 If P is on the opposite side of point B of line m, then B and C are on opposite sisdes of line M and P is on the same side as C. By Lemma 2.7 If P and C are distinct points on the same side of M and ray PC intersects M at point A, then C*P*A or P*C*A. Then with this all true P lies on the ray AC.

(2) By Lemma 2.7 If P and B are on the same side of Line M. If P and B are distinct points on the same side of line M and the ray PB intersects M at point A, then A*P*B or A*B*P. Then with this all true P lies on the ray AB.

Line separation property is independent of axioms I1-3, B1-3 Consider two separate interpretations: (1) the regular Cartesian plane and (2) a new interpretation (Cartesian plane with twist) Points (x,y) Lines ax+by+c=0 A=(0,0) ; P=(1,0) ; B=(2,0) New Between: X°Y°Z : P°A°B whenever X*Y*Z and X not equal to A, P not equal to Y, and Z not equal to B Are all axioms satisfied? Consider point C=(-1,0) is it on line AB, ray AP, Ray AB Question is line AB= ray AP U ray AB
 * B4 - Plane Separation Property** HOMEWORK


 * Theorem 2.12** (Pasch's Theorem) If A,B,C are distinct, noncollinear points and L is any line intersecting AB in a point between A and B, then L also intersects AC or BC. If C does not lie on L, then L does not interect both AC and BC.

A and B are on opposite sides of line L by definition 2.1 If C is on the same side of line L as B then B and C are on same side and BC does not intersect L and A and C on opposite sides by Lemma 2.9 (What is by lemma 2.9? The claim before the reference to it or after?) AC then intersects line L. If B and C are on opposite sides, then by the same logic and Lemma 2.9 BC intersects line L

As a Note, Lines should be designated by lowercase italicized letters for uniformity.