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


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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