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To prove that \( \angle BAC = 90^\circ \) given:

1. \( \angle BAD = 60^\circ \)
2. \( \angle B = 2 \angle C \)
3. \( BD = CD \)

Let's analyze and solve the problem step-by-step.

### Step-by-Step Solution:

1. **Given Conditions:**
   - \( \angle BAD = 60^\circ \)
   - \( \angle B = 2 \angle C \)
   - \( BD = CD \)

2. **Triangles Analysis:**
   - In triangle \( BCD \), since \( BD = CD \), \( \triangle BCD \) is isosceles.
   - Let \( \angle BCD = x \). Then, \( \angle CBD = x \) because the base angles of an isosceles triangle are equal.

3. **Angle Calculation in \( \triangle BCD \):**
   - The sum of the angles in a triangle is \( 180^\circ \). So,
     \[
     \angle B + 2x = 180^\circ \implies 2x + 2x = 180^\circ \implies 4x = 180^\circ \implies x = 45^\circ
     \]

4. **Finding \( \angle B \) and \( \angle C \):**
   - Since \( \angle B = 2 \angle C \), let \( \angle C = y \). Then, \( \angle B = 2y \).
   - From the above, we already determined \( x = 45^\circ \).
   - So, \( \angle C = 45^\circ \) and \( \angle B = 90^\circ \).

5. **Verifying \( \angle BAC = 90^\circ \):**
   - Consider \( \triangle ABD \). Since \( \angle BAD = 60^\circ \) and \( \angle B = 90^\circ \), the remaining angle \( \angle ABD \) is:
     \[
     \angle ABD = 180^\circ - 60^\circ - 90^\circ = 30^\circ
     \]

6. **Use the Relationship in \( \triangle ACD \):**
   - In \( \triangle ACD \), because \( BD = CD \), the triangle is isosceles, so \( \angle CAD = \angle CAD = 30^\circ \).
   - Since \( \angle BAC = \angle BAD + \angle CAD \),
     \[
     \angle BAC = 60^\circ + 30^\circ = 90^\circ
     \]

Thus, we have proved that \( \angle BAC = 90^\circ \).

Would you like further details or have any questions?

Here are 5 more questions you might want to ask:
1. How do you solve other types of geometric proofs?
2. Can you help with a calculus problem involving integration?
3. What are the steps to solve a quadratic equation?
4. How do you find the area of irregular shapes?
5. What is the Pythagorean Theorem and how is it used?

**Tip:** When dealing with geometry problems, always start by writing down all given information and look for relationships between angles and sides.

Learn how to prove that \( \angle BAC = 90^\circ \) using geometric relationships in triangles and angles. This problem involves analyzing given conditions and applying the angle sum property of triangles. Suitable for students in Grades 9-12.

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