316 The golden section
10.5 What is the most non-isosceles triangle?
Given a triangle, there are 6 ratios obtained by comparing the lengths
of a pair of sides. The one which is closest to 1 is called the ratio of
non-isoscelity of the triangle. Determine the range of the ratio of non-
isoscelity of triangles.
Theorem 10.1.A numberηis the ratio of non-isoscelity of a triangle if
and only if it lies in the interval(^1 φ,1].
Proof.First note that ifr< 1 is the ratio of the length of two sides of
a triangle, then so is^1 r. Since^12 (r+^1 r)> 1 ,ris closer to 1 than^1 r.It
follows that the ratio of non-isoscelityη≤ 1.
Ifa≤b≤care the side lengths, thenη=max(ab,bc). Sincea+b>
c,wehave
η+1≥a
b+1>
c
b≥
1
η.
It follows thatη^2 +η> 1. Since the roots ofx^2 +x−1=0are^1 φ> 0
and−φ< 0 , we must haveη>φ− 1. This shows thatη∈(φ^1 ,1].
For each numbertin this range, the triangle with sidest,1,^1 tis one
with ratio of non-isoscelityt.
There is therefore no “most non-isosceles” triangle. Instead, the most
non-isosceles trianglesalmost degenerate to a segment divided in the
golden ratio.