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Circum Radius

O is the circ...

Question

O is the circumcenter of $Δ ABC and R_{1}, R_{2} and R_{3}$ are the radii of the circumcircles of the triangles OBC, OCA and OAB. Then $\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} =$ .

A

\frac{4 abc}{R^{3}}

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B

\frac{8 abc}{R^{3}}

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C

\frac{abc}{R^{3}}

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Solution

The correct option is C $\frac{abc}{R^{3}}$
If O is the circumcenter of $Δ$ ABC, then

OA = OB = OC = R
$Let R_{1}, R_{2} and R_{3} be the circumradii of Δ OBC, Δ OCA and Δ OAB respectively . In Δ OBC, 2 R_{1} = \frac{a}{sin 2 A} or \frac{a}{R_{1}} = 2 sin 2 A or \frac{b}{R_{2}} = 2 sin 2 B and \frac{c}{R_{3}} = 2 sin 2 C \Rightarrow \frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} = 2 (sin 2 A + sin 2 B + sin 2 C) = 8 sinAsinBsinC (∵ sin 2 A + sin 2 B + sin 2 C = 4 sinAsinBsinC) = 8 \frac{a}{2 R} \times \frac{b}{2 R} \times \frac{c}{2 R} = \frac{abc}{R^{3}}$

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