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The problem asks to find the maximum value of \( \cos A \cos B \cos C \) in a plane triangle.

### Solution:
For any triangle, the angles \( A \), \( B \), and \( C \) satisfy the equation:
\[
A + B + C = \pi \quad (\text{since the sum of the angles in a triangle is } 180^\circ).
\]
To maximize \( \cos A \cos B \cos C \), we can take advantage of symmetry and trigonometric identities.

- The function \( \cos x \) reaches its maximum value when \( x \) is as close as possible to \( 0 \), because \( \cos(0) = 1 \) and \( \cos(\frac{\pi}{2}) = 0 \).
- In an equilateral triangle, each angle is \( \frac{\pi}{3} \).

Let's compute:
\[
A = B = C = \frac{\pi}{3}.
\]
Substitute these into the expression:
\[
\cos A \cos B \cos C = \cos\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{3}\right).
\]
Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \), we get:
\[
\cos A \cos B \cos C = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \frac{1}{8}.
\]

### Conclusion:
The maximum value of \( \cos A \cos B \cos C \) in a triangle occurs when the triangle is equilateral, and the value is \( \frac{1}{8} \).

Would you like further details or have any questions?

Here are some related questions:
1. What is the minimum value of \( \cos A \cos B \cos C \)?
2. How does the sum of angles influence the trigonometric functions in a triangle?
3. Can the maximum value be achieved in a non-equilateral triangle?
4. How do angle inequalities affect the product \( \cos A \cos B \cos C \)?
5. What happens to \( \cos A \cos B \cos C \) if one of the angles is obtuse?

**Tip:** In problems involving trigonometric functions in triangles, using symmetry (such as equal angles) often simplifies finding maximum or minimum values.

In a plane triangle, find the maximum value of cos A * cos B * cos C.

Explore the solution to finding the maximum value of cos A * cos B * cos C in a plane triangle. This problem involves trigonometric concepts, angle properties, and maximization techniques. Ideal for students in Grades 10-12.

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