How to prove the directed angle cyclic quadrilateral theorem?"Prove that 4 points A,B,X,Y, no 3 collinear, are concyclic if and only if ∡ X A Y = ∡ X B Y(Where ∡ stands for directed angle mod 180 ∘ )"How does a proof look like that this actually matches the "normal" cyclic quadrilateral theorem?

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How to prove the directed angle cyclic quadrilateral theorem?&quot;Prove that 4 points A,B,X,Y, no 3 collinear, are concyclic if and only if   ∡  X  A  Y  =  ∡  X  B  Y(Where   ∡ stands for directed angle     mod          180    ∘  )&quot;How does a proof look like that this actually matches the &quot;normal&quot; cyclic quadrilateral theorem?

Terry Avery · Accepted Answer

Step 1Let   Φ be a circumcircle of   Δ  A  X  Y  .1. Let B be placed inside   Φ and   B  X  ∩  Φ  =  {  X  ,      B    ′    }.Thus,   ∡  X      B    ′    Y  =  ∡  X  A  Y  =  ∡  X  B  Y  ,, which is a contradiction because   ∡  X  B  Y  &amp;gt;  ∡  X      B    ′    Y  .Step 22. Let B be placed outside   Φ  ..We can get a contradiction by the same way.Id est,   B  ∈  Φ and we are done!

Prove that 4 points A, B, X, Y, no 3 collinear, are concyclic if and only if ∡XAY=∡XBY

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