Maximum area of a polygon inscribed in a complex triangle?

Question

Let $T$ be an acute triangle. Inscribe a pair $R,S$ of rectangles in $T$ as shown :

Let $A(x)$ denote area of polygon $X$ find the maximum value (or show that no maximum exists), of $\frac{A(R)+A(S)}{A(T)}$ where $T$ ranges over all triangles and $R,S$ overall rectangles as above .

Answer 1

Consider the sides as below shown below

As in the figure

$\frac{A(S)+A(R)}{A(T)} = \frac{ay+bz}{\frac{hx}{2}}$

Where $h=a+b+c$ the altitude of $T$. By similar triangles we have,

$\frac{x}{h}=\frac{y}{b+c}=\frac{z}{c}$

So $\frac{A(S)+A(R)}{A(T)} =\frac{2}{h^{2}} (ab+ac+ bc)$

we need to maximise $(ab+bc+ca)$ subject to $a+b+c=h$

One way to do this is to fix $a$ so that $b+c=h-a$

Then , $(ab+bc+ca+)=a(h-a)+bc$

$bc$ is maximised when $b=c$ we now wish to maximise $2ab+b^{2}$ subject to $a+2b=h$ .This is a straightforward calculus problem giving $a=b=c=\frac{1}{3}$

Hence the maximum ratio is $\frac{2}{3}$ ie: independant of $T$.