TITLE: Extreme of $\cos A\cos B\cos C$ in a triangle without calculus.
QUESTION [13 upvotes]: If $A,B,C$ are angles of a triangle, find the extreme value of $\cos A\cos B\cos C$.

I have tried using $A+B+C=\pi$, and applying all and any trig formulas, also AM-GM, but nothing  helps.
On this topic we learned also about Cauchy inequality, but I have no experience with it.
The answer according to Mathematica is when $A=B=C=60$.
Any ideas?

REPLY [2 votes]: We know from geometry that for any triangle ABC the distance between its circumcenter $O$ and its orthocenter $H$ can be given by the following formula:
$$OH^2=R^2(1-8\cos A\cos B\cos C)$$
R being the circumradius.
Besides that, we know that orthocenter and circumcenter coincide only if the triangle is an equilateral one.
Therefore, $\cos A\cos B\cos C$ attains a maximum value of $\frac 18$ when $A=B=C=\pi/3$.
No Calculus needed.