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 $w^2 = \tfrac12$.
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.