# Thread: a:b::b:c when?

1. ## a:b::b:c when?

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

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?

Thanks

2. 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 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 $\displaystyle a:b \:=\: b:c$, then $\displaystyle a,b,c$ form a geometric sequence.

That is, the three sides are: .$\displaystyle a,\;ar,\;ar^2$ in increasing order.
. . WLOG: .$\displaystyle r \ge 1.$

Triangle inequality: . $\displaystyle \begin{array}{ccccc} ar^2 + ar & >& a & [1] \\ ar^2 + a &>& ar & [2] \\ ar + a &>& ar^2 & [3] \end{array}$

Since $\displaystyle r \ge 1$, both [1] and [2] are satsified.

For [3] to be satisfied: .$\displaystyle ar + a \;>\;ar^2$

Divide by $\displaystyle a\!:\;\;r + 1 \;>\;r^2 \quad\Rightarrow\quad r^2-r-1\;<\;0$ .[1]

We have an up-opening parabola: .$\displaystyle y \:=\:x^2 - x - 1$
. . When is it below the $\displaystyle x$-axis?
Answer: between its $\displaystyle x$-intercepts.

The $\displaystyle x$-intercepts are: .$\displaystyle x \;=\;\dfrac{1\pm\sqrt{5}}{2}$

Hence, [1] is satisfied when: .$\displaystyle \dfrac{1-\sqrt{5}}{2} \:<\:r\:<\:\dfrac{1+\sqrt{5}}{2}$

Since $\displaystyle r \ge 1$, we have: .$\displaystyle 1 \;\le\;r\;<\;\dfrac{1+\sqrt{5}}{2}$ .(the Golden Mean)

The triangle must have sides $\displaystyle a,\:ar,\:ar^2$, where $\displaystyle 1 \:\le\:r\:<\:\phi$

3. Thanks. I don't know why but when I learned this ratio I somehow got the impression that it worked for all triangles.