In a triangle ABC, points D and E are taken on side BC such that BD= DE= EC. If angle ADE = angle AED = `theta`, then: (A) tan `theta`= 3 tan B (B) 3 tan `theta` = tan C

Options

A. `tan theta = 3tanB`
B. `3tan theta = tanC`
C. `(6tan theta)/(tan^(2)theta-9) = tanA`
D. `angle B = angleC`

Answer

To solve the problem, we need to analyze the triangle ABC with points D and E on side BC such that BD = DE = EC. Given that angle ADE = angle AED = θ, we will derive the relationships involving the angles and sides of the triangle.

### Step-by-Step Solution:

1. **Understanding the Triangle and Points**:
- Let BD = DE = EC = x. Therefore, BC = BD + DE + EC = x + x + x = 3x.
- Let AD = AE = y (since triangle ADE is isosceles with angles ADE = AED = θ).

2. **Using the Properties of Triangle ADE**:
- In triangle ADE, we can use the tangent function:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{\frac{x}{2}} \]
where h is the height from A to line DE.

3. **Finding the Height (h)**:
- The height h can be expressed in terms of the sides of triangle ABC. Since triangle ABC is not specified, we will express h in terms of the angles and sides we have.
- Using the sine definition, we can express h as:
\[ h = AD \cdot \sin \theta = y \cdot \sin \theta \]

4. **Relating the Angles**:
- In triangle ABC, we can express the tangent of angles B and C in terms of the sides:
\[ \tan B = \frac{h}{BF} \quad \text{and} \quad \tan C = \frac{h}{CF} \]
where BF and CF are segments of BC.

5. **Using the Isosceles Triangle Properties**:
- Since AD = AE, we can say that:
\[ \tan \theta = \frac{h}{\frac{x}{2}} = \frac{2h}{x} \]
- From the previous step, we can substitute h:
\[ \tan \theta = \frac{2y \sin \theta}{x} \]

6. **Finding Relationships**:
- Now, we can express tan B and tan C in terms of θ and x:
\[ \tan B = \frac{h}{BF} = \frac{y \sin \theta}{\frac{3x}{2}} = \frac{2y \sin \theta}{3x} \]
- From the relationship established, we can derive:
\[ \tan \theta = 3 \tan B \]
- Therefore, option (A) is correct:
\[ \tan \theta = 3 \tan B \]

7. **Verifying Option (B)**:
- For option (B), we need to check if \(3 \tan \theta = \tan C\).
- Using similar triangles or the properties of the angles, we can derive that:
\[ 3 \tan \theta \neq \tan C \]
- Therefore, option (B) is not correct.

### Final Answer:
- The correct answer is (A) \( \tan \theta = 3 \tan B \).