Question

The ex-radii of a triangle $r_1,r_2,r_3$ are in H.P, then the sides $a,b,c$ are

• A. in A.P
• B. in G.P
• C. in H.P
• D. in A.G.P

Answer 1

The correct option is A in A.P

$\because r_1,r_2,r_3$ are in H.P.
$\Rightarrow \frac{1}{r_1},\frac{1}{r_2},\frac{1}{r_3}$ are in A.P.
$\therefore \frac{s-a}{△},\frac{s-b}{△},\frac{s-c}{△}$ are in A.P
subtract in each term by $\frac{s}{△}$
Then $\frac{-a}{△},\frac{-b}{△},\frac{-c}{△}$ are in A.P
$\Rightarrow \frac{a}{△},\frac{b}{△},\frac{c}{△}$ are in A.P
$\therefore a,b,c$ are in A.P.