I am trying to solve the problem of finding all triangles with angles $A$, $B$ and $C$ (in $[0,\pi]$) such that $\cos A\cos B+\sin A\sin B\sin C=1$.
In the case where the triangle has a right angle, it can be proved that $C=\frac{\pi}{2}$ (thus $C$ is the right angle and $A=B=\frac{\pi}{4}$. If we assume that the triangle is isosceles, then the triangle is flat or has a right angle (hence is the above solution).
My intuition says that these are the only solutions but I struggle to prove it. Any idea?