AD is internal angle bisector of `DeltaABC " at " angleA` and DE perpendicular to AD which intersects AC at E and meets AB in F, then:

Options

A. `EF=(4bc)/(b+c) sin " A/2`
B. `AD = (2bc)/(b+c) cos " A/2`
C. AE is harmonic mean of b and c
D. `Delta AEF` is an isosceles triangle

Answer

To solve the problem involving triangle \( \Delta ABC \) with the internal angle bisector \( AD \) and the perpendicular \( DE \), we will follow these steps:

### Step 1: Understand the Configuration
We have triangle \( \Delta ABC \) with \( AD \) as the internal angle bisector of \( \angle A \). The line \( DE \) is perpendicular to \( AD \) and intersects \( AC \) at point \( E \) and meets \( AB \) at point \( F \).

**Hint:** Visualize the triangle and the points carefully. Draw the triangle and label the points \( A, B, C, D, E, F \) accordingly.

### Step 2: Use the Angle Bisector Theorem
According to the Angle Bisector Theorem, the ratio of the lengths of the two segments created by the angle bisector \( AD \) on side \( BC \) is equal to the ratio of the lengths of the other two sides of the triangle:
\[ \frac{BD}{DC} = \frac{AB}{AC} \]

**Hint:** Write down the lengths of \( AB \) and \( AC \) if they are given or denote them as \( c \) and \( b \) respectively.

### Step 3: Area Relationships
The area of triangle \( ABC \) can be expressed as the sum of the areas of triangles \( ABD \) and \( ADC \):
\[ \text{Area}(\Delta ABC) = \text{Area}(\Delta ABD) + \text{Area}(\Delta ADC) \]

**Hint:** Remember that the area of a triangle can be calculated using the formula \( \frac{1}{2} \times \text{base} \times \text{height} \).

### Step 4: Determine Lengths Using Perpendicularity
Since \( DE \) is perpendicular to \( AD \), we can use the properties of right triangles to find relationships between the lengths \( EF \), \( AE \), and \( AD \).

**Hint:** Use the Pythagorean theorem in triangles \( ADF \) and \( ADE \) to find the lengths.

### Step 5: Harmonic Mean
To find the harmonic mean of the segments \( AE \) and \( EF \), we use the formula for the harmonic mean:
\[ HM = \frac{2 \cdot AE \cdot EF}{AE + EF} \]

**Hint:** Ensure you have the lengths of \( AE \) and \( EF \) calculated before applying this formula.

### Step 6: Analyze Triangle \( AEF \)
Finally, analyze triangle \( AEF \) to find any additional properties or relationships that may be useful, such as angles or side ratios.

**Hint:** Check if triangle \( AEF \) has any special properties (like being right-angled) that can simplify your calculations.

### Conclusion
By following these steps, we can systematically analyze the triangle and find the required lengths and relationships. Make sure to keep track of all the relationships and properties of the triangles involved.