Question

For a triangle $A B C$, the value of $\cos 2 A+\cos 2 B+\cos 2 C$ is least. If its inradius is $3$ and incentre is $M$, then which of the following is NOT correct?

• A. area of $\triangle A B C$ is $\frac{27 \sqrt{3}}{2}$
• B. $\sin 2 A+\sin 2 B+\sin 2 C=\sin A+\sin B+\sin C$
• C. perimeter of $\triangle A B C$ is $18 \sqrt{3}$
• D. $\overrightarrow{M A} \cdot \overrightarrow{M B}=-18$

Answer 1

Answer: (A)

If $\cos 2 A +\cos 2 B +\cos 2 C$ is minimum then $A =$ $B = C =60^{\circ}$
So $\triangle ABC$ is equilateral
Now in-radias $r=3$
So in $\triangle MBD$ we have
$\operatorname{Tan} 30^{\circ}=\frac{M D}{B D}=\frac{r}{a / 2}=\frac{6}{a}$
$1 / \sqrt{3}=\frac{1}{a}=a=6 \sqrt{3}$
Perimeter of $\triangle ABC =18 \sqrt{3}$
Area of $\triangle ABC =\frac{\sqrt{3}}{4} a^2=27 \sqrt{3}$