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Let's go through the problem step by step:

**Part (1): Determining the shape of triangle \( \triangle ABD \) and proving it.**

From the diagram and the given equation:
\[
AB = BD \cos \angle ABD
\]
This equation suggests a geometric relationship between the sides and angles of triangle \( \triangle ABD \). Specifically, this equation resembles the definition of a cosine function in a right triangle:
\[
\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]
So, \( \triangle ABD \) is likely a right triangle, where \( \angle ABD \) is a right angle (90°). Thus, we can conclude that \( \triangle ABD \) is a right triangle.

**Part (2): Finding the maximum value of diagonal \( AC \) of quadrilateral \( ABCD \).**

We are given:
\[
AB = \sqrt{3} AD, \quad BC = 2\sqrt{3} CD, \quad BC = 12
\]
Let's work through this step-by-step.

1. **Express lengths in terms of variables**: We will first set \( CD = x \), so:
   \[
   BC = 2\sqrt{3} x = 12
   \]
   Solving for \( x \):
   \[
   x = \frac{12}{2\sqrt{3}} = \frac{12}{2} \cdot \frac{1}{\sqrt{3}} = 2\sqrt{3}
   \]
   So, \( CD = 2\sqrt{3} \).

2. **Calculate lengths**:
   Now that we know \( CD = 2\sqrt{3} \), we can substitute into the given relationships. Since \( BC = 12 \), and \( AB = \sqrt{3} AD \), we'll next calculate the length of diagonal \( AC \).

To proceed, we need to set up the relationship between the sides and angles to maximize \( AC \). I'll calculate and simplify this.It seems like I can't do more advanced data analysis right now. Please try again later. However, I can help guide you through the rest of the problem conceptually if you'd like.

Would you like to continue with the explanation, or would you prefer to try solving the next steps manually? Feel free to ask for any clarification on the setup of the equations!

Here are 5 related questions to think about:
1. What geometric properties of right triangles help us verify \( \triangle ABD \) as a right triangle?
2. How would you apply the cosine rule to find the maximum diagonal \( AC \)?
3. What would change if the quadrilateral \( ABCD \) were not planar?
4. How does the relationship between side lengths help us understand the geometry of quadrilateral \( ABCD \)?
5. What other properties can be inferred from the symmetry or angles of \( ABCD \)?

**Tip**: When maximizing a diagonal, think about the angles between the diagonals and how the side lengths interact geometrically.

如图，在平面四边形ABCD中，AB = BD cos∠ABD。
(1)判断△ABD的形状并证明；
(2)若AB=√3AD，BC=2√3CD，BC=12，求四边形ABCD的对角线AC的最大值。

This problem involves determining the shape of triangle ABD using trigonometry and cosine rule, and finding the maximum value of diagonal AC in a quadrilateral ABCD with given relationships between the sides.

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