Problem Set 4.1
4.2 Incidence Theorems Based on the Plane-Separation Postulate
If
E — L = H, U H2 ,
as in the plane-separation postulate, then we say that the sets H 1 and H2 are half planes of L or sides of L. Notice that every line has two sides in every plane that contains it, but if P and Q are on the same side of a line L, this automatically means that L, P, and Q are coplanar. On the other hand, to say that P and Q are on different sides of L, in space, may mean merely that no one plane contains L, P, and Q. If E — L = H1 U H2, as in the plane-separation postulate, then H, and H2 are called opposite sides of L; and if P belongs to H, and Q belongs to H2, we say that P and Q are on opposite sides of L.
The following two theorems are easy.
■ THEOREM I. If P and Q are on opposite sides of the line L, and Q and T are on opposite sides of L, then P and T are on the same side of L.
■ THEOREM 2. If P and Q are on opposite sides of the line L, and Q and T are on the same side of L, then P and T are on opposite sides of L.
We use a similar terminology for the "sides of a point" on a line. That is, if A-B-C, then the rays BA and BC are called opposite rays.
BA BC
• • •
►A
Figure 4.6
■ THEOREM 3. Given a line, and a ray which has its end point on the line but does not lie on the line. Then all points of the ray, except for the end point, are on the same side of the line.
PROOF. Let L be the line, and let AB be the ray, with A E L.
Figure 4.7
4.2 Incidence Theorems Based on the Plane-Separation Postulate 77
Suppose that AB contains a point C such that B and C are on opposite sides of L (in the plane that contains L and AB). Then BC intersects L in some point, and this point must be A, because BC lies in AB , and AB intersects L only in A.
Therefore C-A-B. But this is impossible. By definition, the ray AB is the set of all points C of the line AB for which it is not true that C-A-B. Therefore all points of the ray, other than A, are on the same side of L, namely, the side that contains B. ❑
Similarly for segments:
■
THEOREM 4. Let L be a line, let A be a point of L, and let B be a point not on L. Then all points of AB — A lie on the same side of L.B
A
Figure 4.8
This is true because AB — A lies in AB — A.
Given LBAC.
Figure 4.9
Roughly speaking, the interior of the angle is the set of all points that lie inside it, and the exterior is the set of all points that lie outside it. We can make this idea precise in the following way.
The interior of LBAC is the intersection of the side of AC that contains B, and the side of AB that contains C. Thus a point D lies in the interior (1) if D and B are on the same side of AC, and (2) if D and C are on the same side of AB .
For this definition to be valid, it has to depend only on the angle that we started with, and not on the points B and C that we happened to choose to
L
78 Separation in Planes and Space
describe the angle. Thus, in the figure below, it would be sad if our definition gave us two different interiors for LB'AC' and LBAC:
C C' Figure 4.10
Theorem 3 shows, however, that our definition depends only on the angle, be-cause B and B' are on the same side of AC, and C and C' are on the same side of AB .
Given an angle LAB C, there is exactly one plane E that contains it. The ex-terior of the angle is the set of all points of E that lie neither on the angle nor in its interior.
• THEOREM 5. Every side of a triangle lies, except for its end points, in the interior of the opposite angle.
Here we are using the ordinary terminology; that is, in AABC, the angle LA = LBAC is opposite the side BC.
Figure 4.11
PROOF.
(1) First we apply Theorem 4 to the line AC and the segment BC. By Theorem 4, BC — C lies on the side of AC that contains B.
(2) Next we apply Theorem 4 to the line AB and the segment BC. By Theo-rem 4, BC — B lies on the side of AB that contains C.
(3) By (1) and (2), BC — {B,C} lies in the interior of LBAC. ❑
■
THEOREM 6. If F is in the interior of LBAC, then AF — F lies in the in-terior of LBAC.4.2 Incidence Theorems Based on the Plane-Separation Postulate 79
c Figure 4.12 PROOF.
(1) By definition of the interior of an angle, F and B are on the same side of AC. By Theorem 3, AF — F lies on the side of AC that contains F. There-fore AF — F lies on the side of AC that contains B.
(2) By definition of the interior of an angle, F and C are on the same side of AB. By Theorem 3, AF — F lies on the side of AB that contains F. There-fore AF — F lies on the side of AB that contains C.
By (1) and (2) it follows that AF — F lies in the interior of LBAC. ❑
■
THEOREM7.
Let AABC be a triangle, and let F, D, and G be points, such that B-F-C, A-C-D, and A-F-G. Then G is in the interior of LBCD.C D
Figure 4.13 PROOF.
(1) Since A-F-G, G lies on AF, and A is not between G and F. Therefore G lies on AF. Since G 0 A, G lies on AF — A.
(2) By Theorem 5, F is in the interior of LBAC. It follows by Theorem 6 that AF — A lies in the interior of LBAC. Therefore G and B lie on the same side of AC(= CD).
(3) A and G are on opposite sides of BC, and A and D are on opposite sides of BC. Therefore G and D are on the same side of BC.
By (2) and (3), G is in the interior of LBCD. ❑
Throughout this section, we have been using figures to help us keep track of what is going on. (Here us, of course, includes the author. All authors use
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TIA .80 Separation in Planes and Space
figures, whether or not they show these to the reader.) You should watch closely, however, to be sure that the figures are playing merely their legitimate part as memoranda. It is customary, in elementary texts, for the reader to be assured that "the proofs do not depend on the figure," but these promises are almost never kept. (Whether such promises ought to be kept, in an elementary course, is another question, and the answer should probably be "No.") In a mathemati-cally thorough treatment, however, the hypothesis and conclusion ought to be stated in such a way that no figure is actually necessary to make them plain; and in the same way, the proofs ought to rest on the postulates and the previous theorems. This point is especially relevant in the present context, because in most informal treatments of geometry it is customary to convey betweenness relations and separation properties only by figures, without ever mentioning them in words at all.
You may be able to remember a situation, in elementary geometry, where Theorem 7 is needed.
The interior of /ABC is defined as the intersection of the following three sets:
(1) The side of AB that contains C.
(2) The side of AC that contains B.
(3) The side of BC that contains A.
Figure 4.14
■ THEOREM 8. The interior of a triangle is always a convex set.
Proof?
■ THEOREM 9. The interior of a triangle is the intersection of the interi-ors of its angles.
Proof?