<s>[INST] Mysterious $w$ -- a function of triangle sides Suppose $a$, $b$, $c$ are the lengths of the three sides of a triangle. When studying rational Bézier curves inscribed in the triangle, a mysterious quantity $w$ emerges in the algebra:
$$
w = \frac{c}{\sqrt{2(a^2 + b^2)}}
$$
I'd like to get a better understanding of $w$. Specifically:


*

*Does it represent anything, geometrically? A ratio of lengths or areas, maybe, or some trig function of some angle??

*Can we say anything about its range of values? For example, is it true that $w \le 1$?


A couple of special cases I already figured out: 


*

*If $a=b$ (the triangle is isosceles), then $w$ is the sine of the base angle (the angle between the sides of lengths $a$ and $c$).

*If the sides of length $a$ and $b$ form a right angle, then $w^2 = \tfrac12$.

 [/INST] It looks as if $w^2$ is more tractable. So
$$2w^2=\frac{c^2}{a^2+b^2}$$
and
$$1-2w^2=\frac{a^2+b^2-c^2}{a^2+b^2}=\frac{2ab\cos C}{a^2+b^2}.$$
For fixed $a$ and $b$ as $C$ varies, this varies between $\pm2ab/(a^2+b^2)$. So $2w^2$ varies between $|a-b|^2/(a^2+b^2)$
and $|a+b|^2/(a^2+b^2)$. Thus
$$2w^2\le\frac{a^2+2ab+b^2}{a^2+b^2}\le2$$
by AM/GM so indeed $w\le1$.
I don't see any nice geometry so far.
</s>