JEE Main · 2005 · 08 May 2005 · Offline · Properties of Triangle

With $p_1,p_2,p_3$ the altitudes from $A,B,C$, the area satisfies $\Delta =\frac{1}{2}{p}_{1}a=\frac{1}{2}{p}_{2}b=\frac{1}{2}{p}_{3}c$, so $p_1=\frac{2\Delta }{a}$, $p_2=\frac{2\Delta }{b}$, $p_3=\frac{2\Delta }{c}$. If $p_1,p_2,p_3$ are in H.P., their reciprocals $\frac{a}{2\Delta },\frac{b}{2\Delta },\frac{c}{2\Delta }$ are in A.P., so $a,b,c$ are in A.P. By the sine rule $a,b,c$ are proportional to $\sin A,\sin B,\sin C$, which are therefore also in A.P.