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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.
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 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 points lying on the same side of the line containing the line segment the four points lie on circle for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about 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 covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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.
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 for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about 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 covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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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