Proving Every Line is Contained by Two Planes: Using Incidence Axioms

I-5 states that any two distinct points are contained by at least one line. In summary, to prove that every line is contained by at least two planes using only the incidence axioms, we can use the given information that a line l contains at least two points A and B (conclusion 2), and that space has at least three non-collinear points (conclusion 3). From there, we can use I-1 to show that there exists a line AC and a line BC, which would then form two planes containing line l (conclusion 4). This supports the statement that every line is contained by at least two planes.
  • #1
LCharette
9
0
I need to prove that every line is contained by at least two planes using only the incidence axioms. This is what I have so far...

Conclusions Justifications
1. Let l be any line. Given
2. l has at least two points A and B such that I-5(4)
l = AB.
3. Space has at least 3 non-collinear points, I-5(2)
that is A,B,C
4. There exists line AC and line BC I-1

Help from here?
 
Physics news on Phys.org
  • #2
Please state clearly what "incidence axioms" you are talking about. There are a number of different such sets of axioms, some of which do not HAVE any "planes" at all.
 
  • #3
The incidence axioms regarding lines, points, and planes. For example, I-1 states that any line contains two points.
 

What is an incidence axiom?

An incidence axiom is a fundamental statement in geometry that describes the relationship between points, lines, and planes. It is used to define the basic concepts and properties of these geometric objects.

How many incidence axioms are there?

There are five incidence axioms, which are collectively known as the Euclidean axioms. They were first introduced by the ancient Greek mathematician Euclid in his book "Elements".

What is the purpose of proving incidence axioms?

The purpose of proving incidence axioms is to establish a set of fundamental principles that serve as the basis for further geometric proofs. These axioms are used to construct more complex theorems and to explore the properties of geometric objects.

What are the five incidence axioms?

The five incidence axioms are:
1. The first axiom states that given any two distinct points, there exists a unique line that passes through them.
2. The second axiom states that any line segment can be extended indefinitely to form a line.
3. The third axiom states that given any two distinct lines, there exists at most one point that lies on both lines.
4. The fourth axiom, also known as the parallel postulate, states that if a line intersects two other lines and forms interior angles on the same side that sum to less than 180 degrees, then the two lines will eventually intersect on that side.
5. The fifth axiom, also known as the continuity axiom, states that any two points on a line can be connected by a line segment.

Why are incidence axioms important in geometry?

Incidence axioms are important in geometry because they provide a foundation for understanding and proving geometric concepts. By establishing these fundamental principles, mathematicians can build upon them to develop more complex geometric theories and applications. Additionally, incidence axioms help to define the basic relationships between geometric objects, allowing for a more organized and systematic approach to studying geometry.

Similar threads

I Prove $$T_{p}M$$ is a vector space with the axioms
    • Differential Geometry
Replies
3
Views
1K
MHB Proving Identities Using Axioms of Equality
    • General Math
Replies
16
Views
2K
Trying to prove dual of there are at least tree points on every line
    • Differential Geometry
Replies
11
Views
3K
I EM Power transmitted from one region to another (normal incidence)
    • Electromagnetism
Replies
6
Views
669
B Parametric Representation of Lines
    • Linear and Abstract Algebra
Replies
13
Views
513
Number of lines equidistant from four points on a plane
    • Precalculus Mathematics Homework Help
Replies
2
Views
878
Proving Independence of Fano's Geometry Axiom 4
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
Gmsh is crashing after change Extrude value from 10 to 100
    • Programming and Computer Science
Replies
4
Views
1K
Algorithm problem involving 3 points and 3 lines in the x,y plane
    • Precalculus Mathematics Homework Help
Replies
7
Views
879
Collinear Points -- Ways to determine if points are collinear
    • Precalculus Mathematics Homework Help
Replies
20
Views
1K

Hot Threads

Recent Insights

Back
Top