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Angle Sum Property

If O is cir...

Question

If $O$ is circumcentre of $△ A B C$ and $R_{1}, R_{2}, R_{3}$ are circumradii of triangles $O B C, O C A$ and $O A B$ , then $\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} =$

A

\frac{a b c}{R^{3}}

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B

\frac{2 a b c}{R^{3}}

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C

\frac{a b c}{{2 R}^{3}}

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D

none of these

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Solution

The correct option is A $\frac{a b c}{R^{3}}$
By sine formula in a $△ A B C$
$\frac{a}{sin A} = 2 R \Rightarrow \frac{a}{R} = 2 sin A$ ....(1)
Similarly in $△ O B C, \frac{a}{R_{1}} = 2 sin ∠ B O C;$
in $△ O C A, \frac{b}{R_{2}} = 2 sin ∠ C O A$
and in $△ O A B, \frac{c}{R_{3}} = 2 sin ∠ A O B$
$∴ \frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} = 2 [sin ∠ B O C + sin ∠ C O A + sin ∠ A O B]$
$= 2 [sin 2 A + sin 2 B + sin 2 C] = 2 [4 sin A sin B sin C]$
$= 8 (\frac{a}{2 R}) (\frac{b}{2 R}) (\frac{c}{2 R}) = \frac{a b c}{r^{3}}$

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