Hello, rainer!

Oh my gosh.

I just realized that the famous ratio "a is to b as b is to c" is not always true.

Why would you think that it's true?

Of course it's not true forany three numbers!

This is quite a shocker for me.

Can someone explain what conditions the scalene triangle has to fulfill

for this rule to hold true? .Is it only true for iscoceles? .You meanequilateral.

If , then form ageometric sequence.

That is, the three sides are: . in increasing order.

. . WLOG: .

Triangle inequality: .

Since , both [1] and [2] are satsified.

For [3] to be satisfied: .

Divide by .[1]

We have an up-opening parabola: .

. . When is it below the -axis?

Answer: between its -intercepts.

The -intercepts are: .

Hence, [1] is satisfied when: .

Since , we have: . .(the Golden Mean)

The triangle must have sides , where