Indirect Proof

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Indirect proof definition

Indirect proof in geometry is also called proof by contradiction. The "indirect" part comes from taking what seems to be the opposite stance from the proof's declaration, then trying to prove that. If you "fail" to prove the falsity of the initial proposition, then the statement must be true. You did not prove it directly; you proved it indirectly, by contradiction.

Direct vs. indirect proof

An indirect proof can be thought of as "the long way around" a problem. Rather than attack the problem head-on, as with a direct proof, you go through some other steps to try to prove the exact opposite of the statement. You are subtly intending to fail, so that you can then step back and say, "I did my best to show it was false. I could not prove it was false, so it must be true."

Indirect proof steps

To move through indirect proof logic, you need real confidence and deep content knowledge. The three steps seem simple, much as a one-page cartoon diagram makes assembling furniture seem simple.

Here are the three steps to do an indirect proof:

1. Assume that the statement is false

2. Work hard to prove it is false until you bump into something that simply doesn't work, like a contradiction or a bit of unreality (like having to make a statement that "all circles are triangles," for example)

3. If you find the contradiction to your attempt to prove falsity, then the opposite condition (the original statement) must be true

First step of indirect proof

Geometricians such as yourself can get hung up on the very first step, because you have to word your assumption of falsity carefully.

You first need to clue the reader in on what you are doing. Most mathematicians do that by beginning their proof something like this:

• "Assuming for the sake of contradiction that…"

• "If we momentarily assume the statement is false…"

• "Let us suppose that the statement is false…"

Aha, says the astute reader, we are in for an indirect proof, or a proof by contradiction.

Then you have to make certain you are saying the opposite of the given statement. You cannot say more or less than that for the initial assumption.

Indirect proof examples

Here are three statements lending themselves to indirect proof. Restate each as the beginning of a proof by contradiction:

Given: Two squares

Prove: The two squares are similar figures

Given: An equilateral △ and an angle bisector from any vertex

Prove: The angle bisector is a median

Given: △ABC

Prove: The sum of interior angles of a △ is 180°

Try to come up with the indirect proof statement for each yourself before looking ahead.

In all three cases, begin by presuming the opposite of the statement to be the case:

• "Assume for the sake of contradiction that the two squares are not similar figures…"

• "Let's assume for the moment that the angle bisector of an equilateral △ is not a median…"

• "If we assume the statement is false, then the sum of interior angles of a △ is more or less than 180°"

How to do an indirect proof

When is the right time to try an indirect proof or proof by contradiction? When the statement to be proven true can be questioned: "What if interior angles of triangles do not add to 180°?" Try to prove that; when you fail, you have succeeded!

1. The question to ask is, "What if that statement is not true?"

2. The task to answer is, "How can I prove this statement to be false?"

3. The result should be, "Well, that didn't work, so the original statement has to be true."

Another handy way to use an indirect proof is when the cases showing the statement to be true are simply too numerous to be practical. Consider an assertion like this:

Do you really want to prove that by plugging in every conceivable combination of numbers? Here is a sampling:

Hmmm...that didn't work. Let's try another pair:

You could spend every waking minute plugging in numbers without success.

To solve this using an indirect proof, assume integers do exist that satisfy the equation. Then work the problem:

Given: Where a and b are integers, 10a + 100b = 2

Prove: Integers a and b exist

Divide both sides by 10:

Indirect proof in geometry

Suppose we state this:

Given ∠A and ∠B are supplementary angles:

This is easily proved by indirect proof:

You see the contradiction? ∠B ≥ 180° cannot be greater than or equal to the sum of both angles. Can ∠A be 0°, or even less, a negative?

No; that is not possible. We have proven ∠B < 180° by indirect proof.

Indirect proof, or proof by contradiction, is yet another useful tool to help you with geometry. Use it wisely (it is not suitable for every problem), tell your reader (or teacher) you are using it, and work carefully.