Question

Let $ABC$ be the triangle with $AB=1,AC=3$ and $\angle BAC=\frac{\pi }{2}$. If a circle of radius $r>0$ touches the sides $AB,AC$ and also touches internally the circumcircle of the triangle $ABC$, then the value of $r$ ___

Answer 1

Answer: 0.84

$C_1\left(\frac{1}{2},\frac{3}{2}\right)$ and $r_1=\frac{\sqrt{10}}{2}$
$C_2=(r,r)$
$\therefore$ circle $C_2$ touches $C_1$ internally
$\Rightarrow C_1{C}_2=\left|r-\frac{\sqrt{10}}{2}\right|$
$\Rightarrow {\left(r-\frac{1}{2}\right)}^2+{\left(r-\frac{3}{2}\right)}^2={\left(r-\frac{\sqrt{10}}{2}\right)}^2$
$r^2-4r+\sqrt{10}r=0$
$r=0\text{ (reject) or }r=4-\sqrt{10}$