In a $△ A B C, r_{A}, r_{B}, r_{C}$ are the radii of the circles which touch the incircle and the sides emanating from the vertices $A, B, C$ respectively,
then $\sqrt{r_{A} r_{B}} + \sqrt{r_{B} r_{C}} + \sqrt{r_{C} r_{A}} = ?$

r

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2 r

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\frac{r}{2}

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None of these

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Solution

The correct option is A $r$
Let the circle of radius $r_{A}$ touch the sides $A B$ and $A C$ at $D$ and $E,$
whereas the incircle touches these sides at $D$ and $E .$

Let $O$ and $O^{'}$ be the centres of the inscribed and that of the circle with radius $r_{A} .$
$O$ and $O^{'}$ lie on the bisector of angle $A .$

Also, $A O^{'} = O D^{'} csc \frac{A}{2} = r_{A} csc \frac{A}{2}$
$A O = O D csc \frac{A}{2} = r csc \frac{A}{2}$

Hence; $O O^{'} = r + r_{A} = A O - A O^{'}$

$\Rightarrow r + r_{A} = r (csc \frac{A}{2}) - r_{A} (csc \frac{A}{2})$

$∴ \frac{r_{A}}{r} = \frac{csc \frac{A}{2} - 1}{csc \frac{A}{2} + 1} \Rightarrow r_{A} = r {tan}^{2} (\frac{π - A}{4})$

Similarly, $r_{B} = r {tan}^{2} \frac{π - B}{4}, r_{C} = r {tan}^{2} \frac{π - C}{4}$

Hence, $\sqrt{r_{A} r_{B}} + \sqrt{r_{B} r_{C}} + \sqrt{r_{C} r_{A}}$

$= r {tan \frac{π - A}{4} . tan \frac{π - B}{4} + tan \frac{π - B}{4} . tan \frac{π - C}{4} + tan \frac{π - C}{4} . tan \frac{π - A}{4}}$

$= \frac{r}{cos \frac{π - A}{4} cos \frac{π - B}{4} . cos \frac{π - C}{4}}$ ${\sum sin \frac{π - A}{4} sin \frac{π - B}{4} cos \frac{π - C}{4}}$

$= \frac{r}{π cos \frac{π - A}{4}} \times$ $[sin \frac{π - A}{4} (sin \frac{π - B}{4} . cos \frac{π - C}{4} + sin \frac{π - C}{4} . cos \frac{π - B}{4})$
$+ sin \frac{π - B}{4} . sin \frac{π - C}{4} . cos \frac{π - A}{4}]$

$= \frac{r}{π cos \frac{π - A}{4}} \times [cos \frac{π + A}{4} . cos \frac{π - A}{4} + sin \frac{π - B}{4} . sin \frac{π - C}{4} . cos \frac{π - A}{4}]$

$= \frac{r cos \frac{π - A}{4}}{cos \frac{π - A}{4} . cos \frac{π - B}{4} . cos \frac{π - C}{4}}$ $= [cos \frac{π - B + π - C}{4} + sin \frac{π - B}{4} . sin \frac{π - C}{4}]$

$= \frac{r cos \frac{π - A}{4} . cos \frac{π - B}{4} . cos \frac{π - C}{4}}{cos \frac{π - A}{4} . cos \frac{π - B}{4} cos \frac{π - C}{4}} = r$

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In a △ABC,rA,rB,rC are the radii of the circles which touch the incircle and the sides emanating from the vertices A,B,C respectively, then √rArB+√rBrC+√rCrA=?

In a $△ A B C, r_{A}, r_{B}, r_{C}$ are the radii of the circles which touch the incircle and the sides emanating from the vertices $A, B, C$ respectively,
then $\sqrt{r_{A} r_{B}} + \sqrt{r_{B} r_{C}} + \sqrt{r_{C} r_{A}} = ?$