In a `!ABC` bisector of angle C meets the side AB at D and circumcricle at E. The maximum value of CD . DE is equal to

Options

A. `(b^(2))/4`
B. `(c^(2))/4`
C. `(a^(2))/4`
D. none of these

Explanation

To solve the problem, we need to find the maximum value of the product \( CD \cdot DE \) where \( D \) is the point where the angle bisector of \( \angle C \) meets side \( AB \) and \( E \) is the point where the angle bisector intersects the circumcircle of triangle \( ABC \).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
We have triangle \( ABC \) with angle \( C \). The angle bisector of \( \angle C \) intersects side \( AB \) at point \( D \) and the circumcircle of triangle \( ABC \) at point \( E \).

2. **Using the Power of a Point Theorem**:
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