Prove that any point on a circle forms a right-angled triangle with any diameter of this circle.

To prove that any point on a circle forms a right-angled triangle with any diameter of the circle, we can use the **Thales' theorem**. Here's the proof:

### Theorem:
**Thales' Theorem** states that if \( A \), \( B \), and \( C \) are points on a circle where \( AC \) is a diameter, then the angle \( \angle ABC \) is a right angle (90°).

### Proof:

1. **Consider a circle** with center \( O \) and a diameter \( AC \). Let \( B \) be any point on the circle that is not on the line \( AC \).

2. Since \( AC \) is the diameter, \( OA = OC = r \) where \( r \) is the radius of the circle.

3. The triangle \( OAB \) and \( OBC \) are both isosceles triangles since \( OA = OB = r \) and \( OB = OC = r \).

4. Let's denote the angle \( \angle OAB = \theta_1 \) and \( \angle OBC = \theta_2 \).

5. Now, consider the angles \( \angle OAC \) and \( \angle OCA \), which are the same because \( OA = OC \). Each of these angles is \( 90^\circ - \theta_1 \) and \( 90^\circ - \theta_2 \) respectively.

6. The angle \( \angle ABC \) is an exterior angle for triangle \( OAC \), so:
   \[
   \angle ABC = \theta_1 + \theta_2
   \]

7. Since the sum of angles in triangle \( ABC \) is \( 180^\circ \) and \( \theta_1 + \theta_2 \) is equal to the right angle \( \angle OBC \):
   \[
   \theta_1 + \theta_2 = 90^\circ
   \]
   Hence, \( \angle ABC = 90^\circ \).

### Conclusion:
Since \( \angle ABC = 90^\circ \), any point \( B \) on the circle forms a right-angled triangle with the endpoints \( A \) and \( C \) of any diameter \( AC \) of the circle. This proves Thales' theorem.

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Would you like more details on any part of this proof, or do you have any questions?

### Related Questions:
1. How is Thales' Theorem applied in real-world problems?
2. What are other important theorems related to circles in geometry?
3. Can you explain the converse of Thales' Theorem?
4. How would the proof

Explore the proof of Thales' theorem that demonstrates how any point on a circle creates a right-angled triangle with any diameter. This geometric problem uses Thales' theorem, which is fundamental in circle geometry and proofs involving triangles. Suitable for high school students.

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