Can Cotangent Sums Exceed 2/3 in Acute Triangles with Perpendicular Medians?

[Albert1]

point $G$ is the barycenter of an acute triangle $\triangle ABC$ ,if $\overline{BG}\perp \overline{CG}$
prove $cot\,\, B +cot\,\, C\geq \dfrac {2}{3}$

 

[Albert1]

point $G$ is the barycenter of an acute triangle $\triangle ABC$ ,if $\overline{BG}\perp \overline{CG}$
prove $cot\,\, B +cot\,\, C\geq \dfrac {2}{3}$

hint :

Spoiler

construct points $M,\,\,and \,\,H\,\, on \,\,\overline {BC}$
where $M$ is the midpoint of $\overline {BC}$ and $\overline{AH}\perp \overline {BC}$

 

[Albert1]

hint :

Spoiler

construct points $M,\,\,and \,\,H\,\, on \,\,\overline {BC}$
where $M$ is the midpoint of $\overline {BC}$ and $\overline{AH}\perp \overline {BC}$

solution:

Spoiler

View attachment 4639