Geometry/Proof

Introduction
Unlike science which has theories, mathematics has a definite notion of proof. Mathematics applies deductive reasoning to create a series of logical statements which show that one thing implies another.

Consider a triangle, which we define as a shape with three vertices joined by three lines. We know that we can arbitrarily pick some point on a page, and make that into a vertex. We repeat that process and pick a second point. Using a ruler, we can connect these two points. We now make a third point, not on the line through the first two points, and using the ruler connect it to each of the other points. We have constructed a triangle.

In mathematics we formalize this process into axioms, and carefully lay out the sequence of statements to show what follows. All definitions are clearly defined. In modern mathematics, we are always working within some system where various axioms hold.

Two-Column Proof
The most common form of explicit proof in high school geometry is a two column proof consists of five parts: the given, the proposition, the statement column, the reason column, and the diagram (if one is given).

Example of a Two-Column Proof
Now, suppose a problem tells you to solve $$x + 1 = 2$$ for $$x$$, showing all steps made to get to the answer. A proof shows how this is done:

Given: $$x + 1 = 2$$

Prove: x = 1

We use "Given" as the first reason, because it is "given" to us in the problem.

Written Proof
Written proofs (also known as informal proofs, paragraph proofs, or 'plans for proof') are written in paragraph form. Other than this formatting difference, they are similar to two-column proofs.

Sometimes it is helpful to start with a written proof, before formalizing the proof in two-column form. If you're having trouble putting your proof into two column form, try "talking it out" in a written proof first.

Example of a Written Proof
We are given that x + 1 = 2, so if we subtract one from each side of the equation (x + 1 - 1 = 2 - 1), then we can see that x = 1 by the definition of subtraction.

Flowchart Proof
A flowchart proof or more simply a flow proof is a graphical representation of a two-column proof. Each set of statement and reasons are recorded in a box and then arrows are drawn from one step to another. This method shows how different ideas come together to formulate the proof.