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 for any 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 mean equilateral.
If , then form a geometric sequence.
That is, the three sides are: . in increasing order.
. . WLOG: .
Triangle inequality: .
Since , both  and  are satsified.
For  to be satisfied: .
Divide by .
We have an up-opening parabola: .
. . When is it below the -axis?
Answer: between its -intercepts.
The -intercepts are: .
Hence,  is satisfied when: .
Since , we have: . .(the Golden Mean)
The triangle must have sides , where