Question

If area of a triangle is 2 sq. units, then find the value of the product of the arithmetic mean of the lengths of the sides of a triangle and harmonic mean of the lengths of the altitudes of the triangle.

Answer 1

Let `h_(1), h_(2), and h_(3)` be the lengths of alititudes throguh vertices A, B, and C, respectively
`a h_(1) = b h_(2) = c h_(3) = 2 Delta`
or `(1)/(h_(1)) + (1)/(h_(2)) + (1)/(h_(3)) = (a + b + c)/(2 Delta)`
or `(a + b + c)/(3) xx (3)/((1)/(h_(1)) + (1)/(h_(2)) + (1)/(h_(3))) = 2 Delta = 4`