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Standard Values of Trigonometric Ratios

If O is the...

Question

If $O$ is the circumcentre of $Δ A B C$ and $R_{1}, R_{2}$ and $R_{3}$ are the radii of the circumcircles of triangles $O B C, O C A$ and $O A B$ , respectively, then $\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}}$ has the value equal to

A

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

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B

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

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C

\frac{4 Δ}{R^{2}}

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D

\frac{Δ}{4 R^{2}}

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Solution

The correct options are
A $\frac{a b c}{R^{3}}$
D $\frac{4 Δ}{R^{2}}$
If O is the circumcentre of $Δ A B C$ , then $O A = O B = O C$ Let $R_{1}, R_{2}$ and $R_{3}$ be circumradii of $Δ O B C, Δ O C A$ and $Δ O A B$ respectively.
In $Δ O B C, 2 R_{1} = \frac{a}{s i n 2 A} \Rightarrow \frac{a}{R_{1}} = 2 s i n 2 A$
Similarly $\frac{b}{R_{2}} = 2 s i n 2 B$ and $\frac{c}{R_{3}} = 2 s i n 2 C$
$\Rightarrow \frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} = 2 (s i n 2 A + s i n 2 B + s i n 2 C) = 8 s i n A s i n B s i n C$
$= 8 \frac{a}{2 R} \frac{b}{2 R} \frac{C}{2 R} = \frac{a b c}{R^{3}}$

This is also equal to $\frac{4 Δ}{R^{2}}$

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