If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on the circle (i.e. they are concyclic).

The statement you're asking about relates to a fundamental property of circles in geometry. When a line segment connects two points and subtends equal angles at two other points on the same side of the line, it implies that all four points are concyclic, meaning they lie on the same circle.

Understanding the Concept

To break this down:

• Line Segment: This is the straight path connecting two points, say A and B.
• Subtending Angles: The angles formed at points C and D (which are not on the line segment) by the endpoints A and B.
• Equal Angles: If angle ACB equals angle ADB, then the points A, B, C, and D are concyclic.

Why This Happens

This property arises from the Inscribed Angle Theorem, which states that an angle inscribed in a circle is half of the central angle that subtends the same arc. Therefore, if two angles subtended by the same arc are equal, the points must lie on the same circle.

Visualizing the Points

Imagine drawing a circle that passes through points A, B, C, and D. The angles at C and D, formed by the line segment AB, will always be equal if the conditions are met. This geometric relationship is crucial in various proofs and constructions in circle geometry.

In summary, if you have two points on a line segment that subtend equal angles at two other points, you can confidently say that all four points lie on a single circle. This principle is a key aspect of circle theorems in geometry.