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

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In an acute triangle ABC with barycenter G, if the medians BG and CG are perpendicular, it is necessary to prove that the sum of the cotangents of angles B and C is at least 2/3. The discussion revolves around geometric properties and relationships between the angles and medians. Participants explore various mathematical approaches and theorems to establish this inequality. The focus remains on the implications of the perpendicularity condition on the cotangent values. Ultimately, the proof aims to confirm the stated inequality under the given conditions.
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point $G$ is the barycenter of an acute triangle $\triangle ABC$ ,if $\overline{BG}\perp \overline{CG}$
prove $cot\,\, B +cot\,\, C\geq \dfrac {2}{3}$
 
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Albert said:
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 :
construct points $M,\,\,and \,\,H\,\, on \,\,\overline {BC}$
where $M$ is the midpoint of $\overline {BC}$ and $\overline{AH}\perp \overline {BC}$
 
Albert said:
hint :
construct points $M,\,\,and \,\,H\,\, on \,\,\overline {BC}$
where $M$ is the midpoint of $\overline {BC}$ and $\overline{AH}\perp \overline {BC}$
solution:

 

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