Properties of Triangles 2 Question 8
8. Prove that a $\triangle A B C$ is equilateral if and only if $\tan A+\tan B+\tan C=3 \sqrt{3}$.
(1998, 8M)
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Solution:
- If the triangle is equilateral, then
$$ A=B=C=60^{\circ} $$
$\Rightarrow \tan A+\tan B+\tan C=3 \tan 60^{\circ}=3 \sqrt{3}$
Conversely assume that,
$$ \tan A+\tan B+\tan C=3 \sqrt{3} $$
But in $\triangle A B C, A+B=180^{\circ}-C$
Taking tan on both sides, we get
$$ \tan (A+B)=\tan \left(180^{\circ}-C\right) $$
$$ \Rightarrow \quad \frac{\tan A+\tan B}{1-\tan A \tan B}=-\tan C $$
$\Rightarrow \tan A+\tan B=-\tan C+\tan A \tan B \tan C$
$\Rightarrow \tan A+\tan B+\tan C=\tan A \tan B \tan C=3 \sqrt{3}$
$\Rightarrow$ None of the $\tan A, \tan B, \tan C$ can be negative
So, $\triangle A B C$ cannot be obtuse angle triangle.
Also, $\quad AM \geq GM$
$\Rightarrow \frac{1}{3}[\tan A+\tan B+\tan C] \geq[\tan A \tan B \tan C]^{1 / 3}$
$\Rightarrow \quad \frac{1}{3}(3 \sqrt{3}) \geq(3 \sqrt{3})^{1 / 3} \Rightarrow \sqrt{3} \geq \sqrt{3}$.
So, equality can hold if and only if $\tan A=\tan B=\tan C$
or $A=B=C$ or when the triangle is equilateral.
