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Geometry Problem 764: Exploring the Relationship Between Triangle, Inradius, Altitudes, and Harmonic Mean - Perfect for High School and College Students!

The figure shows a triangle ABC with the inradius r and the altitudes ha, hb, hc. Prove that the inradius is one-third the harmonic mean of the altitudes, therefore, \(\dfrac{1}{r}=\dfrac{1}{h_{a}}+\dfrac{1}{h_{b}}+\dfrac{1}{h_{c}}\)  

Triangle, Altitudes, Inradius, Harmonic Mean
 

Triangle's inradius
One-third harmonic mean of
Heights, beauty revealed.

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