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If a line segment joining two point subtends equal angles at two other points lying on the same side of the line containing the line segment the four points lie on circle
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If a line segment joining two point subtends equal angles at two other...
Understanding the Theorem
The theorem states that if a line segment connecting two points, say A and B, subtends equal angles at two other points, say C and D, which lie on the same side of the line containing segment AB, then points A, B, C, and D are concyclic, meaning they lie on the circumference of a single circle.
Key Concepts
- Equal Angles: The angles ∠ACB and ∠ADB are equal. This is a crucial condition that leads to the conclusion of concyclicity.
Geometric Visualization
- Circle Definition: A circle is defined as the locus of all points equidistant from a center point. In this scenario, points A, B, C, and D lie on the circumference of a circle.
- Inscribed Angle Theorem: The theorem states that an angle subtended by an arc at any point on the circle is half of the angle subtended at the center. Since ∠ACB = ∠ADB, both angles subtend the same arc AB.
Proof Outline
- Constructing the Circle: Draw the line segment AB. Since ∠ACB = ∠ADB, construct the circle with points A and B as endpoints of the arc containing points C and D.
- Cyclic Quadrilateral: The quadrilateral formed by points A, B, C, and D is cyclic, meaning it can be inscribed in a circle.
Conclusion
This theorem beautifully illustrates the relationship between angles and circles in geometry. By understanding this concept, one can easily determine the position of points in relation to a circle, enhancing spatial reasoning and geometric problem-solving skills.
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If a line segment joining two point subtends equal angles at two other...
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If a line segment joining two point subtends equal angles at two other points lying on the same side of the line containing the line segment the four points lie on circle
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