For a triangle ABC, the value of [ 2A + 2B + 2C] is least.

Question

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

Accepted Answer

If $\cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C}$ is minimum then $\mathrm{A}=$

$\mathrm{B}=\mathrm{C}=60^{\circ}$

So $\triangle \mathrm{ABC}$ is equilateral

Now in-radias $r=3$

So in $\triangle \mathrm{MBD}$ we have

$$
\begin{aligned}
& \operatorname{tan} 30^{\circ}=\frac{M D}{B D}=\frac{r}{a / 2}=\frac{6}{a} \\
& 1 / \sqrt{3}=\frac{1}{a}=a=6 \sqrt{3}
\end{aligned}
$$

Perimeter of $\triangle \mathrm{ABC}=18 \sqrt{3}$

Area of $\triangle \mathrm{ABC}=\frac{\sqrt{3}}{4} a^2=27 \sqrt{3}$

Solutions · 1