# How do you proof that for #a,b,cinRR#, #a/b=b/c=c/a<=>a=b=c#?

##### 2 Answers

The reverse direction is trivial, assuming we add the condition that

For the forward direction, note first that all of

Now, there are two possible cases in which the claim would not hold. In the first case, one differs from the other two. Without loss of generality, suppose

In the second case, they all differ. Then, without loss of generality, suppose

Thus the only way to avoid a contradiction is if

Here's an alternative proof not using proof by contradiction...

#### Explanation:

Let me try to prove this without using contradiction:

First, if

To show in the other direction, suppose

Then

So we find:

#a = bk = ck^2 = ak^3#

Dividing both ends by

#k^3 = 1#

Since we are told that

So

Notice my comment above about the Real cube root. If we were not told that

For example,