The statement provided is correct. When an interval (line segment) subtends equal angles at two points on the same side of it, the end points of the interval and the two points are concyclic, meaning they lie on the same circle.
To understand this concept, consider the following steps:
1. Start with an interval AB, where A and B are the endpoints.
2. Choose two points, P and Q, on the same side of AB.
3. If the angles APB and AQB are equal, it means that the lines AP and AQ or BP and BQ intersect at the same angle.
4. According to the inscribed angle theorem, if two angles in the same segment are equal, then the points defining those angles lie on the same circle.
5. Applying the inscribed angle theorem to the angles APB and AQB, we can conclude that the points A, B, P, and Q are concyclic, meaning they lie on the same circle.
In simpler terms, if an interval (line segment) subtends equal angles at two points on the same side of it, it means that these four points lie on a circle.
For example, consider a line segment AB with points P and Q on the same side. If the angles APB and AQB are equal, then points A, B, P, and Q are concyclic and lie on the same circle.
This property is useful in various geometric proofs and constructions, allowing us to establish relationships and connections between points and angles on a circle.