<s>[INST] $a,b,c$ are the sides and $A,B,C$ are the angles of a triangle. If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal then, 
$a,b,c$ are the sides of a $\triangle ABC$ and $A,B,C$ are the respective angles. If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal then $\sin^2 \bigl(\frac{A}{2}\bigr), \sin^2 \bigl(\frac{B}{2}\bigr)$ and $\sin^2\bigl(\frac{C}{2}\bigr)$ are in which type of progression?(Arithmetic,geometric,harmonic)

My try, 
I immediately noticed that $1$ was a root and since the product of the roots was $1$ I got that $a,b,c$ were in harmonic progression. I replaced the reciprocals of $a,b,c$ by $p,q,r$. Wrote $\sin^2\bigl({\theta\over 2}\bigr)$ as $\frac{1-cos(\theta)}{2}$ and replaced $\cos$ by the sides but did not succeed. I also tried by replacing it with $\frac{(s-b)(s-c)}{bc}$ but still could not get the required answer.
If I assume the result I am getting the answer by simplifying the expression.
Please give an elegant solution. 
 [/INST] As you noticed, the roots are,
$$x = 1\hspace{1 cm}\mathrm{ or }\hspace{1 cm}x = \frac{(a-b)c}{a(b-c)}$$
The roots are equal,$$1 = \frac{(a-b)c}{a(b-c)}.$$
Therefore, we can find
$$b = \frac{2ac}{a+c}.$$
Now let's talk about trigonometry. The half angle formula for sine is
$$\sin^2{\frac{W}{2}} = \frac{1 - \cos{W}}{2}$$
and the cosine law is
$$w^2 = u^2 + v^2 - 2 u v \cos{W}$$
Therefore,
$$-\frac{\cos{W}}{2} = \frac{w^2 - u^2 - v^2}{4 u v}$$
Therefore,
$$\sin^2{\frac{W}{2}} = \frac{1}{2} + \frac{w^2 - u^2 - v^2}{4 u v} = \frac{w^2 - (u - v)^2}{4uv}.$$
Now we can find that
$$\sin^2{\frac{A}{2}}=\frac{\left(a^2+2 a c-c^2\right) \left(a^2+c^2\right)}{8 a c^2 (a+c)}$$
$$\sin^2{\frac{B}{2}}=\frac{1}{4} \left(-\frac{a}{c}+\frac{4 a c}{(a+c)^2}-\frac{c}{a}+2\right)$$
$$\sin^2{\frac{C}{2}}=\frac{-a^4+2 a^3 c+2 a c^3+c^4}{8 a^2 c (a+c)}$$
We then just need to check the progression types.
</s>