12'1 Lines and Planes in Space

1
12.1 Lines and Planes in Space
2
The Three Axioms

• There is a unique plane containing three given
  noncollinear points. If A, B, and C are the given
  points, then we will denote the one plane they
  determine by p(A, B, C).
• If two distinct points are in a plane, then the
  entire line they determine also is in the plane.
• If two planes intersect, then their intersection
  consists of more than one point.

3
Lemma

• If l is a line and P is a point not on l, then
  there is a unique plane p that contains P and l.
• If l1 and l2 are distinct intersecting lines,
  then there is a unique plane p which contains
  both l1 and l2.
• There is at most one plane containing two
  distinct lines.

4
Theorem

• If two distinct planes intersect, then their
  intersection is a line.

5
Theorem

• Let l1 and l2 be distinct lines in the plane p
  that intersect at the point F. If P is a point
  that is not on p with and
  then is perpendicular to any line in p that
  passes through F.

P
D
A
X
Y
F
B
C
6
A
Proof
D
X
Y
F
B
C

• Given
• And by vertical angle theorem
  
• Then by SAS
• Therefore and
  ( by vertical angles)
• And by ASA
• This implies FXFY and AXCY.

7
Proof continued
P
A
F
C

• We know
• And
  
• So that
  by SAS.
• This implies PAPBPCPD.
• So that by SSS.

8
Proof continued
P
P
A
X
B
C
Y
D

• Then so that
  by SAS.
• This implies PXPY and
  by SSS.
• Therefore
• Where
  . So that

P
X
Y
F
9
Corollary

• Let F be a point on a plane p and suppose points
  P1 and P2 are not on p . If and
  then .
• Let F1 and F2 be points on a plane p and suppose
  point P is not on p . If and
  then .

10
Theorem

• Let be perpendicular to the plane p at F,
  P a point not on p, and suppose points A and B
  lie in p. Then FAltFB if and only if PAltPB.

P
F
A
B
11
Theorem
Corollary

• Two distinct lines each perpendicular to the same
  plane are parallel.

• If lines l1 and l2 are parallel and l1 is
  perpendicular to the plane p, then .
• If l1, l2 , and l3 are lines such that l1// l2
  and l2// l3, then l1// l3.

12
Theorem

• If P is a point not on the plane p, then there is
  a unique line l through P such that
• .

P
1.3 by SSS
M
A
B
13
Lemma

• Let p 1 and p 2 be parallel planes.
• If p is a plane that intersects p 1 in l1 and p 2
  in l2, then l1// l2.
• If a line , then .

p 1
p 1
p
p 2
14
Theorem

• If P1 and Q1 are points on the plane p1, and P2
  and Q2 are points on the plane p2, and p1//p2,
  then dist(P1, p2) dist(Q1, p2) dist(P2, p1)
  dist(Q2, p1).

p 1
P1
Q1
p 2
P2
Q2
15
14.1

• Theorem
• If a plane p intersects a sphere with center O
  and radius r a distance x dist(O, p) from the
  center, where Oxlt r, then the intersection is a
  circle with radius .
• Corollary
• A unique great circle is determined by any two
  points of a sphere that are not antipodes.
• Corollary
• Three points on a sphere determine a unique
  circle of the sphere.

16
Homework Problem

• Prove the first Theorem in 12.1 If two distinct
  planes intersect, then their intersection is a
  line.

Reference

• Berele, Allan and Goldman, Jerry. Geometry
  Theorems and Constructions. Prentice Hall, New
  Jersey 2001.