To prove that four points lie on the same circle (are concyclic), we use the converse of the inscribed angle theorem.
Converse of the Inscribed Angle Theorem
If two points P and Q on the same side of two points A and B satisfy ∠APB = ∠AQB, then the four points A, P, Q, and B lie on the same circle.
📘 Example ①
Given four points A, B, C, and D, if ∠ADB = ∠ACB, state why the four points lie on the same circle.
Answer: Since ∠ADB = ∠ACB for the same arc AB (the inscribed angles are equal), the converse of the inscribed angle theorem shows that the four points A, D, C, and B lie on the same circle.
Given four points A, B, C, and D, if ∠ADB = ∠ACB, state why the four points lie on the same circle.
Answer: Since ∠ADB = ∠ACB for the same arc AB (the inscribed angles are equal), the converse of the inscribed angle theorem shows that the four points A, D, C, and B lie on the same circle.
💡 Key Point
- Equal angles subtending the same arc → four points are concyclic
- Use this to prove that four points lie on the same circle
Practice Problems
- When ∠APB = ∠AQB = 70°, state the positional relationship among A, P, Q, and B.
Answers and Explanations
- Answer: The four points A, P, Q, and B lie on the same circle (they are concyclic).