To solve this problem, let's analyze the given information step-by-step.

Problem Details

1. Inscribed Quadrilateral $ABCD$:

   • Quadrilateral $ABCD$ is inscribed in a circle, meaning all its vertices lie on a common circle.

2. Diagonal $AC$ is a Diameter:

   • This implies that $\angle ABC$ and $\angle ADC$ are right angles, as any angle subtended by a diameter in a circle is a right angle.

3. Equal Lengths $AB = BD$:

   • The sides $AB$ and $BD$ are equal in length.

4. Radius of Circumscribed Circle is 6:

   • This gives us that the radius $R = 6$, so the diameter $AC = 2 \times R = 12$.

5. Distance from $C$ to the Intersection of Diagonals is 2:

   • Let $O$ be the point of intersection of the diagonals $AC$ and $BD$. It is given that $CO = 2$.

Objective

We are asked to find the cosine of the angle $\angle ABD$.

Solution Steps

1. Use the Right Angles: Since $\angle ABC = 90^\circ$ and $\angle ADC = 90^\circ$ (as $AC$ is a diameter), triangles $ABC$ and $ADC$ are right triangles.

2. Applying the Power of a Point Theorem: The Power of a Point theorem states that for two intersecting chords $AB$ and $CD$ in a circle, the product of the segments is equal. Here: $AO \cdot OC = BO \cdot OD$ Since $AB = BD$, $BO = OD$.

3. Using Given Values:

   • With $CO = 2$ and knowing $AC = 12$, we can find $AO$: $AO = AC - OC = 12 - 2 = 10.$

4. Calculate $\cos \angle ABD$: Using trigonometry in right triangles $ABC$ or $ABD$, we could find $\cos \angle ABD$ if additional values for other sides were known.

Conclusion

Currently, there is insufficient information to complete the solution without additional values, particularly for the length of $AB$ or $BD$. With these, $\cos \angle ABD$ can be found.

Would you like assistance in exploring possible methods or need clarification on any specific steps?

Practice Questions

1. How would the solution change if $CO$ were different?
2. What if $AB \neq BD$ but instead $AB + BD = AC$?
3. How does the Power of a Point theorem apply in quadrilaterals?
4. Why does $\angle ABC = 90^\circ$ if $AC$ is a diameter?
5. How can we determine side lengths if only angles are given?

Tip

For problems involving cyclic quadrilaterals, remember that opposite angles are supplementary. This property often simplifies trigonometric calculations.