What does this figure signify?

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.

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$