Portal:Euclidean geometry/Chapter 3

As explained in chapter 2, conditional statements are any statement that can take the form "if p, then q".

This same principle applies to making logical arguments in proofs, the topic of this chapter.

Introduction

In chapter 2, you learned the basics of how to create a proof. However, it would be proper here to "dissect" such proof in order to understand its parts.

Part	Example
1. Theorem.	a2 + b2 = c2
2. Logical reasoning with wit and, possibly, other postulates and theorems.	Proof by diagram
3. Restate theorem. (Q.E.D.)	a2 + b2 = c2

The goal of this chapter is to explain how to complete part two.

The Middle Third

The first two thirds of a proof are easy: state the theorem, restate the theorem. The middle third is the trickier part, and the part we have to focus on. The first thing one must do if trying to complete the middle third is to state any given information or postulates. This is your basis for any logical argument.

Second, one must infer from the given information. For example, if we are given a polygon with four sides, knowing that two of them are parallel and equal in length, and asked to prove that it is a parallelogram, then we have to conclude that because the figure is a polygon, the four sides are connected, making the other two sides parallel. When we know that all four sides are parallel, the figure is a parallelogram by definition. Q.E.D.

The Alternative Method

There is one other way to go about solving a proof. The way to start this method is to negate (see chapter 2) your theorem and show how this negation would lead to a contradiction in terms. Once you've done this, the only logical solution is that the theorem is true, and Q.E.D, you've proven what you needed to prove.

Why Proofs are Important

Proofs are able to prove things outside of the field of mathematics. If enough information is given to you, the proof of anything is merely a derivative, and therefore very much possible.