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

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SUMMARY

The discussion centers on the geometric properties of acute triangles, specifically focusing on the barycenter point \( G \) and the relationship between the cotangents of angles \( B \) and \( C \). It is established that if the medians \( \overline{BG} \) and \( \overline{CG} \) are perpendicular, then the inequality \( \cot B + \cot C \geq \frac{2}{3} \) holds true. This conclusion is derived from the properties of acute triangles and the behavior of cotangent functions in relation to angle measures.

PREREQUISITES
  • Understanding of acute triangles and their properties
  • Knowledge of barycenters in triangle geometry
  • Familiarity with cotangent functions and trigonometric identities
  • Basic principles of median lines in triangles
NEXT STEPS
  • Study the properties of barycenters in various types of triangles
  • Explore trigonometric inequalities involving cotangent functions
  • Investigate the implications of perpendicular medians in triangle geometry
  • Learn about advanced geometric proofs involving acute triangles
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Mathematicians, geometry enthusiasts, and students studying triangle properties and trigonometric functions will benefit from this discussion.

Albert1
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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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