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- Feb 14, 2012
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Let ABC be a triangle. Prove that $sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}+2sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=1$.
Conversely, prove that if x, y and z are positive real numbers such that $x^2y^2+z^2+2xyz=1$, then there is a triangle ABC such that $x=sin\frac{A}{2}, y=sin\frac{B}{2}, z=sin\frac{C}{2}$.
I can solve the first part of the question.
My problem is I can't see a way to proceed with the second half of the question.
They are closely-related, I know, but I just couldn't see the right way to go.
Any help/hint would be deeply appreciated.
Thanks.
Conversely, prove that if x, y and z are positive real numbers such that $x^2y^2+z^2+2xyz=1$, then there is a triangle ABC such that $x=sin\frac{A}{2}, y=sin\frac{B}{2}, z=sin\frac{C}{2}$.
I can solve the first part of the question.
My problem is I can't see a way to proceed with the second half of the question.
They are closely-related, I know, but I just couldn't see the right way to go.

Any help/hint would be deeply appreciated.
Thanks.