Let $A, B, C$ be angles of a triangle and $a,b,c$ be the respective opposite sides.

Problem: If $\sin^2 \frac{A}{2},\sin^2 \frac{B}{2},\sin^2 \frac{C}{2}$ are in HP(harmonic progression), prove that $a,b,c$ are in HP too.

Solution. We have,
$$\frac{s(s-a)}{(s-b)(s-c)},\frac{s(s-b)}{(s-a)(s-c)},\frac{s(s-c)}{(s-a)(s-b)}$$ are in AP(arithmetic progression).
Writing the condition for arithmetic mean,
$$\frac{2s(s-b)}{(s-a)(s-c)}=\frac{s(s-a)}{(s-b)(s-c)}+\frac{s(s-c)}{(s-a)(s-b)}$$
$$\Rightarrow 2(s-b)^2=(s-a)^2+(s-c)^2$$
Substituting $s=\frac{a+b+c}{2}$ and simplifying yields,
$$2ac=bc+ab$$
$$\Rightarrow b=\frac{2ac}{a+c}$$
Hence, $a,b,c$ are in HP. (end of proof)

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