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

Raelynn Nolan

Raelynn Nolan

Open question

2022-08-22

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?

Answer & Explanation

Terry Avery

Terry Avery

Beginner2022-08-23Added 10 answers

Step 1
Let 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 > X B Y .
Step 2
2. Let B be placed outside Φ ..
We can get a contradiction by the same way.
Id est, B Φ and we are done!

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