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 h_{a}, h_{b}, h_{c}. Prove that the inradius is one-third the
harmonic mean of the altitudes, therefore,
\(\dfrac{1}{r}=\dfrac{1}{h_{a}}+\dfrac{1}{h_{b}}+\dfrac{1}{h_{c}}\)