Simple Geometry

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Its given that $AD=32 , ~~~CD=40 , ~~~BE=34$

Now as tangents from same external point are equal

$\Rightarrow \therefore AD=AE~~~CD=CF~~~BE=BF$

Now area of $\triangle ABC$ By herons formula

First of all we'll find $s$ i.e semi perimeter

$\Rightarrow s=\dfrac { 66+72+74 }{ 2 } =106\\ \Rightarrow \therefore \left[ ABC \right] =\dfrac { \sqrt { 106(40)(34)(32) } }{ 1 } =32\sqrt { 4505 }$

Now we have formula for inradius (denoted by $r$ )

$\Rightarrow r=\dfrac { 2\cdot A }{ a+b+c }$

Here $A$ =Area of triangle

$\Rightarrow \therefore r=\dfrac { 2\cdot 32\sqrt { 4505 } }{ 66+72+74 } \\ \Rightarrow r=\dfrac { 2\cdot 32\sqrt { 4505 } }{ 212 } \\ \Rightarrow r=\dfrac { 32\sqrt { 4505 } }{ 106 } \\ \Rightarrow r\approx 20.2624$

$\therefore$ just greater integer than the answer is $\huge\Rightarrow\color{royalblue}{\boxed{21}}$

Note that we can find the sides of the triangle in this way:

$a=32+34=66 \quad\quad b=32+40=72 \quad\quad c=34+40=74$

Now, calculate the area using Heron's formula:

$A=\dfrac{\sqrt{(66+72+74)(66+72-74)(66-72+74)(-66+72+74)}}{4}$

$A=32\sqrt{4505}$

Finally, compute the inradius with the formula:

$r=\dfrac{2A}{a+b+c}$

$r=\dfrac{64\sqrt{4505}}{66+72+74}=\dfrac{16\sqrt{4505}}{53}$

$r \approx 20.2624 \implies \lceil r \rceil =\boxed{21}$

First of all notice that the length of the sides is NOT given .

Now to the solving part .

Let the incircle touch the side AB at P where AP = 32 , Let I be the incentre and r be the inradius . Then AI bisects < BAC .

Therefore from the right angled $\Delta$ IPA , $\frac{r}{32}$ = $\tan \frac{A}{2}$ .

Therefore, 32= $r \cot \frac{A}{2}$ . Similarly , 34= $r \cot \frac{B}{2}$ and 40= $r \cot \frac{C}{2}$ .

In $\Delta$ ABC , we have the identity $\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2}$

= $\cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2}$

Therefore ,
$\frac{32}{r} + \frac{34}{r} + \frac{40}{r}$ = $\frac{32}{r} . \frac{34}{r} . \frac{40}{r}$

After simplifying ,

$r = 20.26$

Now the answer is the smallest integer greater than 20.26 which is 21 .