Question

If 'O' is the circum centre of the $\Delta ABC$ and $R_1,R_2$ and $R_3$ are the radii of the circumcircles of triangles OBC, OCA and OAB respectively then $\frac{a}{R_1}+\frac{b}{R_2}+\frac{c}{R_3}$ has the value equal to

• A. $\frac{abc}{2R^3}$
• B. $\frac{R^3}{abc}$
• C. $\frac{4\Delta }{R^2}$
• D. $\frac{abc}{R^3}$

Answer 1

Answer: (C), (D)

$\frac{abc}{R^3}$
$\frac{4\Delta }{R^2}$

$ar\left(\Delta OBC\right)+ar\left(\Delta OAC\right)+ar\left(\Delta OAB\right)=ar\left(\Delta ABC\right)$

$\Rightarrow \frac{a{R}^2}{4R_1}+\frac{b{R}^2}{4R_2}+\frac{c{R}^2}{4R_3}=\frac{abc}{4R}$

$\Rightarrow \frac{a}{R_1}+\frac{b}{R_2}+\frac{c}{R_3}=\frac{abc}{R^3}=\frac{4\Delta }{R^2}$

Hence, options C and D.