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The problem asks you to demonstrate that an angle inscribed in a semicircle is a right angle. Specifically, it's focused on a circle where the angle \(\theta\) is inscribed, and two of its vertices lie on the diameter of the circle, at points \(A(-r, 0)\) and \(B(r, 0)\), with a third point \(C(x, y)\) on the circle.

The key idea to solving this problem is that any angle inscribed in a semicircle (with its base along the diameter) is a right angle (90°), a fact based on Thales' theorem.

### Steps:

1. **Equation of the Circle**:
   The circle's equation is given by:
   \[
   x^2 + y^2 = r^2
   \]
   where \(r\) is the radius of the circle.

2. **Vectors of the Triangle**:
   You're asked to express the sides as vectors. Let’s define the vectors of the two sides of the triangle \(AC\) and \(BC\), with points \(A(-r, 0)\), \(B(r, 0)\), and \(C(x, y)\):
   
   - Vector \(\overrightarrow{AC}\): 
     \[
     \overrightarrow{AC} = (x - (-r), y - 0) = (x + r, y)
     \]
   - Vector \(\overrightarrow{BC}\): 
     \[
     \overrightarrow{BC} = (x - r, y - 0) = (x - r, y)
     \]

3. **Dot Product and Perpendicularity**:
   To show that the angle \(\theta\) is a right angle, we need to check if the two vectors are perpendicular. Vectors are perpendicular if their dot product is zero.

   The dot product of \(\overrightarrow{AC}\) and \(\overrightarrow{BC}\) is:
   \[
   \overrightarrow{AC} \cdot \overrightarrow{BC} = (x + r)(x - r) + y \cdot y
   \]
   Simplifying this:
   \[
   (x + r)(x - r) + y^2 = x^2 - r^2 + y^2
   \]
   Since the point \(C(x, y)\) lies on the circle, we know that \(x^2 + y^2 = r^2\). Substituting this into the equation:
   \[
   x^2 - r^2 + y^2 = r^2 - r^2 = 0
   \]
   Therefore, the dot product is zero, meaning the two vectors \(\overrightarrow{AC}\) and \(\overrightarrow{BC}\) are perpendicular, and the angle \(\theta\) is a right angle.

### Conclusion:
The inscribed angle \(\theta\) formed by the diameter of the circle is a right angle (90°), as shown by the perpendicularity of the vectors.

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Do you need further clarification or more details?

Here are 5 related questions to expand your understanding:
1. How does Thales' theorem apply to any point on a circle's circumference?
2. Can you find the angle between two vectors using the cross product method?
3. What is the relationship between a circle's radius and the points on its circumference?
4. How can we generalize this result to any triangle inscribed in a semicircle?
5. How would this proof change if the circle were not centered at the origin?

**Tip:** When using vectors, always check if the dot product equals zero to confirm perpendicularity, which is crucial in proving right angles!

Montrez qu'un angle inscrit sur le diamètre d'un cercle est un angle droit. Rappel: L'équation d'un cercle de rayon r est donnée par x² + y² = r². Indice: Exprimez les côtés en rouge comme des vecteurs.

This solution demonstrates how to prove that an angle inscribed in a semicircle is a right angle using the dot product of vectors. Using the equation of a circle x² + y² = r², the problem is solved step-by-step, showcasing key vector geometry principles. Suitable for high school students.

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