In `Delta ABC`, I is the incentre, Area of `DeltaIBC, DeltaIAC and DeltaIAB` are, respectively, `Delta_(1), Delta_(2) and Delta_(3)`. If the values of `Delta_(1), Delta_(2) and Delta_(3)` are in A.P., then the altitudes of the `DeltaABC` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Answer

C

Explanation

`Delta_(1) = (1)/(2) ra, Delta_(2) = (1)/(2) rb, Delta_(3) = (1)/(2) rc`
Given that `Delta_(1), Delta_(2), Delta_(3)` are in A.P.
Therefore, a, b, c are in A.P.
Let `Delta` be the area of triangle ABC. Also, let x, y and z be the measures of the altitudes of the triangle
Then `x = (2Delta)/(a), y = (2Delta)/(b), z = (2Delta)/(c)`
Hence, x,y, z are in H.P.