Question

In a $\Delta ABC$, $AB=AC$ and the length of the altitude from $A$ to $BC$ is $h$. The area and perimeter of $\Delta ABC$ is $A$ and $P$ respectively. The circumradius of $\Delta ABC$ is $r$. The value of $\frac{1}{r\underset{h\to 0}{lim}\frac{A}{P^3}}$ is

Answer 1

In $\Delta BOD$,
$B{D}^2=r^2-(h-r{)}^2\Rightarrow BD=\sqrt{2hr-h^2}$
In $\Delta BAD$,
$A{B}^2=h^2+(2hr-h^2)\Rightarrow AC=AB=\sqrt{2hr}$

$P=2(\sqrt{2hr-h^2}+\sqrt{2hr})$
$A=\frac{1}{2}\times [2\left(\sqrt{2hr-h^2}\right)h]$
$\Rightarrow A=h\sqrt{2hr-h^2}$

$\underset{h\to 0}{lim}\frac{A}{P^3}=\underset{h\to 0}{lim}\frac{h\sqrt{2hr-h^2}}{8h^{3/2}(\sqrt{2r-h}+\sqrt{2r}{)}^3}$
$=\frac{1}{8}\frac{\sqrt{2r}}{(\sqrt{2r}+\sqrt{2r}{)}^3}$
$=\frac{1}{8}\frac{\sqrt{2r}}{8\left(2r\right)\sqrt{2r}}$
$\underset{h\to 0}{lim}\frac{A}{P^3}=\frac{1}{128r}$
$\Rightarrow \frac{1}{r\underset{h\to 0}{lim}\frac{A}{P^3}}=128$