What does a geometric proof look like?

Question

High school student here. I've never really had a chance to prove anything considering my current educational standing, but I am very interested in mathematics and would like a to try to prove some very basic things in the near future. Recently, I've learned that there are different types of proofs, specifically geometric ones. I'm wondering what this would look like and if this approach can be used on any problem.

Answer 1

As I learned high school geometry decades ago, it is the first time you see axioms and have to derive things more formally. You are given the axioms for Euclidean geometry, which typically sound obvious because they agree with our experience. In class various theorems are proven, often in a two column format. The left column has statements and the right column has justifications, which can be algebraic or can be citations to axioms or theorems that have already been proven. They are often accompanied by a diagram showing what is going on.

An example would be to prove the base angles of an isosceles triangle are equal. You define an isosceles triangle as one having two equal sides. The proof would show a diagram with a line cutting the triangle in half from the vertex between the equal sides like the one below. The small lines cutting the two sides indicate they are equal. You would specify that $CH$ is a median, that it connects that vertex to the midpoint of the opposite side. Presumably you have shown you can construct this point. The proof would be something like $$\begin {array} {l l}CH=CH&\text{Same segment}\\AC=BC&\text{Given}\\AH=HB&\text{By construction} \\ACH \cong BCH&\text{side-side-side}\\ \angle A=\angle B&\text{corresponding parts of congruent triangles}\end {array}$$

Answer 2

Well, first thing you need to understand is that the "geometric" adjective is not a well-defined thing when applied to proofs. What it refers to, loosely speaking, is a proof that is significantly easier to comprehend if accompanied by a particular picture (whether the picture is included in the proof, or left to the readers to construct for themselves).

For example, the classic Basel Problem can be attacked from many different angles, but here is a nice geometric way of looking at the problem.

A word of caution though: a picture by itself is rarely considered proof by itself. There typically needs to be some text to contextualise it, and explain how this picture deals with the result in full generality.