Thales' theorem is a fundamental result in geometry that states:

**If \( A \), \( B \), and \( C \) are points on a circle where \( AB \) is a diameter of the circle, then the angle \( \angle ACB \) is a right angle (90°).**

### Explanation:

- Consider a circle with a diameter \( AB \).
- Let \( C \) be any point on the circumference of the circle (but not on the diameter \( AB \)).
- Then, the angle \( \angle ACB \) formed by drawing lines \( AC \) and \( BC \) to the endpoints of the diameter is always a right angle.

### Proof Outline:
1. **Let \( O \) be the center of the circle**, making \( OA = OB = OC = r \) (where \( r \) is the radius).
2. **Triangle \( OAC \) and triangle \( OBC \)** are isosceles because \( OA = OC \) and \( OB = OC \).
3. The angles at the center are \( \angle AOC \) and \( \angle BOC \).
4. \( \angle AOC + \angle BOC = 180^\circ \) since \( AB \) is a straight line through \( O \).
5. Since \( \angle AOC = \angle BOC \), each of these angles is \( 90^\circ \), making \( \angle ACB = 90^\circ \).

### Applications:
- Thales' theorem is widely used in constructing perpendicular lines and understanding the properties of circles.
- It also forms

Explore Thales' theorem, a fundamental result in geometry stating that if points A, B, and C lie on a circle with AB as a diameter, then angle ACB is a right angle (90°). Learn its proof and applications in constructing perpendicular lines and circle properties. Suitable for Grades 9-12.

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