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Geometry Problem 764: Triangle, Inradius, Altitudes, Harmonic Mean. Level: High School, College.

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
 

 

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