Question about a proof of the converse of Thales' theorem

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Homework Statement
Proof of the converse of Thales' theorem
Relevant Equations
(x–u)(x–v) = 0
I was watching this series of videos of Eddie Woo proving Thales' theorem and its converse. I didn't understand this part (at 2:15) where he considered (x–u)(x–v) = 0. He later used the result he got from considering that. But why consider it in the first place?

 

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