I made this after I saw a similar problem in our module, where we were given above defined \(\omega\) and I developed on the circumcircles using properties of circles.

We had to prove -

\(\cot(A) + \cot(B) + \cot(C) = \cot(\omega)\)

and

\(\csc^2(\omega) = \csc^2(A) + \csc^2(B) + \csc^2(C)\)

Can we take any help from this circles help in proving the above identities?

I joined each center with \(O\) to obtain an isosceles triangle and something like -

\((\cot(\omega) - \cot(A))(\cot(\omega) - \cot(B))(\cot(\omega) - \cot(C)) = \frac{1}{\sin(A)\sin(B)\sin(C)}\)

I am missing something.

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