Mathematics - Properties of Triangle Question with Solution | TestHub

We know the identity $\cot A+\cot B+\cot C=\frac{a^2+b^2+c^2}{4\Delta }$, where $\Delta$ is the area of the triangle. For an equilateral triangle, $A=B=C=60^{\circ }$. In this case, $\cot A=\cot B=\cot C=\cot 60^{\circ }=\frac{1}{\sqrt{3}}$. So, $\cot A+\cot B+\cot C=\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}}=\frac{3}{\sqrt{3}}=\sqrt{3}$.

It is a known property that for an acute-angled triangle, the minimum value of $\cot A+\cot B+\cot C$ is $\sqrt{3}$, which occurs when the triangle is equilateral. If the triangle is obtuse, the sum would be greater than $\sqrt{3}$. Thus, the condition $\cot A+\cot B+\cot C=\sqrt{3}$ implies that the triangle must be equilateral.