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I serach for my question But I didn't find same question. my question is:
if we have Inscribed quadrilateral, then we can easily prove that $\alpha = \beta $

enter image description here

But If we know that $\alpha = \beta $, then how can we prove that our quadrilateral is Inscribed in a circle?
I do something for this but I am not sure about it. can you verify it?

we know that $\alpha = \beta $ but we suppose that our quadrilateral ABCD is not Inscribed. Then we make ABCF that is Inscribed quadrilateral so because $\alpha = \beta $ we have $\smallfrown EC = \smallfrown CF$ (I don't know how to make it) and It's not true so our quadrilateral ABCD is Inscribed.

I'm sorry for bad English. Thanks. enter image description here

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This is (the converse of) the Inscribed Angle Theorem.

Given a segment $\overline{PQ}$, the locus of points $X$ such that $\angle PXQ$ is a given size is an arc of a circle through $P$ and $Q$.

Thales' Theorem is the particular case when $\angle PXQ$ must be a right angle. In that case, points $X$ lie on the circle with diameter $\overline{PQ}$.

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