Question

The ex - radii $r_1,r_2,r_3$ of $\Delta$ ABC are in H.P . Show that its sides a, b, c are in A.P

Answer 1

Since $r_1,r_2,r_3$ are in H.P. it follows that
or $\frac{s-a}{S},\frac{s-b}{S},\frac{s-c}{S}$ are in A.P.
Hence $\frac{2(s-b)}{S}=\frac{s-a}{S}+\frac{s-c}{S}$
or -2b = -a - c or b = $\frac{1}{2}$ (a + c)
Hence a,b,c are in A.P.