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In the given diagram, we have a circle with center \( O \) and radius \( r \), and points \( A \), \( B \), and \( C \) lie on the circumference of the circle. \( O \) is the center of the circle, and \( AB \), \( AC \), and \( BC \) are chords of the circle. The goal is to explain why angle \( \angle ABC \) is a right angle.

### Explanation using **Thales' Theorem**:

This configuration suggests the use of **Thales' Theorem**, which states:

- *If \( A \), \( B \), and \( C \) are points on a circle such that \( AC \) is a diameter of the circle, then the angle \( \angle ABC \) formed by the points is a right angle.*

Let's break this down:

1. **The Diameter**: In the diagram, \( A \), \( C \), and \( O \) are collinear, with \( AO = r \) and \( CO = r \), meaning that the line \( AC \) is the diameter of the circle (since the diameter is twice the radius). Therefore, \( AC \) passes through the center \( O \).

2. **Thales' Theorem Application**: According to Thales' Theorem, any triangle formed by the endpoints of a diameter (here \( A \) and \( C \)) and a point on the circle (here \( B \)) will have a right angle at that point on the circumference. Since \( AC \) is a diameter and \( B \) lies on the circumference, the angle \( \angle ABC \) must be a right angle.

Thus, the angle \( \angle ABC \) is a right angle because it subtends the diameter \( AC \) of the circle, in accordance with Thales' Theorem.

Would you like more details or clarifications? Here are 5 related questions to deepen your understanding:

1. How can we formally prove Thales' Theorem?
2. What are the properties of a triangle inscribed in a semicircle?
3. Can Thales' Theorem apply to quadrilaterals or other polygons?
4. How can we generalize Thales' Theorem for non-right triangles?
5. How do the radii \( r \) influence the angles and side lengths in this triangle?

**Tip**: In problems involving circles and angles, identifying whether a chord is a diameter is crucial, as it often leads to applications of Thales' Theorem.

This problem demonstrates how to prove that angle ABC is a right angle using Thales' Theorem. The diagram involves a circle with center O and a triangle inscribed such that AC is the diameter, making angle ABC a right angle.

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