How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - \cot A)$?

Question

Consider all triangles $ABC$ where $A < B < C \leq \frac{\pi}{2}$. How find the least value of the expression:

$M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - \cot A)$?

Answer 1

consider $(\cot B - \cot C)(\cot C - \cot A)=-(\cot C-\cot B )(\cot C - \cot A)$

for easy, $x=\cot C,a=\cot A,b=\cot B,ab=-(a+b)x+1 ,f(x)=x^2-(a+b)x+ab=x^2-2(a+b)x+1$ to have max

$x< (a+b) \implies f_{max}=f(0)=1$

$\cot^2 C \ge 0$ so $M$ will get least value when $\cot C=0$

rest is easy.

indeed, without the limitation of $A,B,C$. $M$ will also have least value $1$ but there is one more case is $A=B=C$