If x sin theta = y sin left( theta + frac{2pi | vedclass

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(B) Let $x \sin 	heta = y \sin \left( 	heta + \frac{2\pi}{3} ight) = z \sin \left( 	heta + \frac{4\pi}{3} ight) = k$.Then $x = \frac{k}{\sin

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$xy + yz + zx = 0$

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(B) Let $x \sin 	heta = y \sin \left( 	heta + \frac{2\pi}{3} ight) = z \sin \left( 	heta + \frac{4\pi}{3} ight) = k$.Then $x = \frac{k}{\sin 	heta}$,$y = \frac{k}{\sin(	heta + 2\pi/3)}$,and $z = \frac{k}{\sin(	heta + 4\pi/3)}$.Consider the sum $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{k} [\sin 	heta + \sin(	heta + 2\pi/3) + \sin(	heta + 4\pi/3)]$.Using the identity $\sin A + \sin(A + 120^\circ) + \sin(A + 240^\circ) = 0$,we get $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$.This simplifies to $\frac{yz + zx + xy}{xyz} = 0$,which implies $xy + yz + zx = 0$.

If $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right)$,then:

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