Question

In a triangle $ABC$, the incircle touches the sides $BC,CA$ and $AB$ at $D,E,F$ respectively.

If the radius of incircle is $4$ units and $BD,CE$ and $AF$ be consecutive natural numbers, then the sum of digits of smallest length is?

Answer 1

Let BD, CE and AF be of lengths $y-1$, $y$ and $y+1$ respectively.
Since the lengths of the tangent, from an external point to the circle are equal
$BF=BD=y-1,CD=CE=y$ and $AE=AF=y+1$
$\Rightarrow BC=2y-1,CA=2y+1$ and $AB=2y$,
$\Rightarrow s=3y$
Now, $\tan \frac{C}{2}=\frac{r}{s-c}$
$\Rightarrow \tan \frac{C}{2}=\frac{r}{y}$
$\tan \frac{A}{2}=\frac{r}{s-a}=\frac{r}{y+1}$
and $\tan \frac{B}{2}=\frac{r}{s-b}=\frac{r}{y-1}$
$\tan (\frac{B}{2}+\frac{C}{2})=\frac{\frac{r}{y-1}+\frac{r}{y}}{1-\frac{r^2}{y(y-1)}}$
$\Rightarrow \cot \frac{A}{2}=\frac{2ry-r}{y^2-y-r^2}$
$\Rightarrow \frac{y+1}{r}=\frac{2ry-r}{y^2-y-r^2}$
$\Rightarrow y^3-3r^{2}y-y=0$
$\Rightarrow y^3-48y-y=0$ [as r $=$ 4 (given)].
$\Rightarrow y=0$ or $y^2=49\Rightarrow y=7$ (as $y\ne 0$)
So, the sides of the triangle are 13, 14 and 15 units.
Required sum $1+3=4$