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In $\Delta ABC$, if $a, b, c$ are in $A.P.$ (with usual notations), identify the incorrect statements -
$h_1, h_2, h_3$ are in $H.P.$, where $h_1, h_2, h_3$ are altitudes from vertices $A,B$ $C$ respectively.
$sinA, sinB, sinC$ are in $A.P.$
$r_1, r_2, r_3$ are in $A.P.$
$tan \frac{A}{2} , tan \frac{B}{2}, tan \frac{C}{2} $ are in $H.P.$
Solution
$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}.$
$\lambda \mathrm{a}, \lambda \mathrm{b}, \lambda \mathrm{c}$ will be in $\mathrm{AP}.$
$\Rightarrow \sin A, \sin B, \sin C$ in $A P.$
Also $\mathrm{h}_{1}=\frac{2 \Delta}{\mathrm{a}} ; \mathrm{h}_{2}=\frac{2 \Delta}{\mathrm{b}} ; \mathrm{h}_{3}=\frac{2 \Delta}{\mathrm{c}}$ will be in $HP.$
$a, b, c$ in $A P \Rightarrow s-a, s-b, s-c$ in $A P$
$\Rightarrow \frac{\Delta}{s-a}, \frac{\Delta}{s-b} ; \frac{\Delta}{s-c}$ will be in $HP.$
$\Rightarrow \mathrm{r}_{1}, \mathrm{r}_{2}, \mathrm{r}_{3}$ in $H.P.$
Also $\frac{\Delta}{s(s-a)}, \frac{\Delta}{s(s-b)}, \frac{\Delta}{s(s-c)}$ will be in $HP.$
