Q:

The question of the geometric sequence

Let $h_a$, $h_b$,$h_c$ are heights of a triangle with $h_b^2=h_a.h_c$
and $q = \frac{{{h_b}}}{{{h_a}}}$.
Which of the numbers 2 and 3 and 1.9 and 1.5  can be value of $q$?

A:

Let's see. The sides of a triangle are inversely proportional to its heights, so they too form a geometric sequence with the same ratio. Also, the sides are subject to the triangle inequality: $a<b+c$. Since $a=q^2\cdot c$ and $b=q\cdot c$, this essentially means $q^2<q+1$, or $q<\varphi$ (where $\varphi$ is the golden ratio). So 1.5 would do, but 1.9 and larger values wouldn't.

