To solve the problem, we can use the following steps:

1. Understand the given information:

   • Triangle $ABC$ is isosceles with $AB = AC$.
   • $\angle ABC = \angle ACB = 2 \times \angle D$.
   • Tangents to the circle at points $B$ and $C$ meet at point $D$.

2. Apply the property of tangents and the circle:

   • The angles formed between the tangent and the chord through the point of contact (e.g., $\angle DBC$ and $\angle DCB$) are equal to the angles in the alternate segments.
   • Therefore, $\angle DBC = \angle DAC$ and $\angle DCB = \angle DAB$.

3. Express $\angle A$ using the known angles:

   • Since $ABC$ is isosceles, $\angle ABC = \angle ACB = 2 \times \angle D$.
   • Let $\angle D = x$. Therefore, $\angle ABC = \angle ACB = 2x$.
   • The sum of angles in triangle $ABC$ is $180^\circ$, so: $\angle BAC + 2x + 2x = 180^\circ$ $\angle BAC + 4x = 180^\circ$ $\angle BAC = 180^\circ - 4x$

4. Determine the angle $\angle A$:

   • The angle $\angle A$ in triangle $ABC$ is given by $\angle BAC = 180^\circ - 4x$.

5. Conclusion:

   • Since $x = \angle D$, the final expression for $\angle A$ is: $\angle A = 180^\circ - 4 \times \angle D$

Thus, angle A is $180^\circ - 4 \times \angle D$.

Would you like a detailed explanation on any specific step or have any further questions?

Here are five related questions:

1. How does the tangent-segment theorem apply to this problem?
2. How would the solution change if triangle ABC was equilateral?
3. What would be the effect on the angles if $\angle D$ was given a specific value?
4. Can this method be generalized to other types of triangles inscribed in a circle?
5. How do you prove that the tangents at B and C are equal in length?

Tip: Remember that the sum of angles in any triangle is always $180^\circ$, a key fact often used in solving geometric problems.