Question

${BP}^2+{CQ}^2+{AR}^2=$

• A. $4R^2[1-2\cos A\cos B\cos C]$
• B. $4R^2[1+2\cos A\cos B\cos C]$
• C. $8R^2\cos A\cos B\cos C$
• D. $4R^2[1+2\sin A\sin B\sin C]$

Answer 1

The correct option is A $4R^2[1-2\cos A\cos B\cos C]$

$AP=2R,{BP}^2={AP}^2-{AB}^2=4R^2-c^2=4R^2-4R^2{\sin }^{2}C=4R^2{\cos }^{2}C$

Similarly ${CQ}^2=4R^2{\cos }^{2}A$ and ${AR}^2=4R^2{\cos }^{2}B$

$\therefore {BP}^2+{CQ}^2+A{R}^2=4R^2({\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C)$

$=4R^2\{\frac{3}{2}+\frac{1}{2}(\cos 2A+\cos 2B+\cos 2C)\}$

$=4R^2\{\frac{3}{2}-\frac{1}{2}-2\cos A\cos B\cos C\}=4R^2[1-2\cos A\cos B\cos C]$