ABC is a given triangle. AD, BE and CF are altitudes of `DeltaABC`.
Assertion (A) : `(AB^(2)+BC^(2)+CA^(2)) gt (AD^(2)+BE^(2)+CF^(2))`
Reason (R) : `(AE^(2)-AF^(2))+(BF^(2)-BD^(2))+(CD^(2)-CE^(2))=0`

Options

A. Both Assertion A and Reason R are correct and Reason R is a correct explanation of Assertion A.
B. Both Assertion A and Reason R are correct but Reason R is not the correct explanation of Assertion A.
C. Assertion A is correct, Reason R is wrong.
D. Assertion A is wrong, Reason R is correct.

Answer

To solve the assertion and reason question regarding triangle ABC and its altitudes AD, BE, and CF, we will analyze both statements step by step.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
The assertion states that \( AB^2 + BC^2 + CA^2 > AD^2 + BE^2 + CF^2 \). This is a known property in geometry where the sum of the squares of the sides of a triangle is always greater than the sum of the squares of the altitudes.

**Hint**: Recall the property of triangles that relates the sides and altitudes.

2. **Analyzing the Reason**:
The reason given is \( AE^2 - AF^2 + BF^2 - BD^2 + CD^2 - CE^2 = 0 \). This equation suggests a relationship between segments created by the altitudes and points on the sides of the triangle.

**Hint**: Check if the segments can be rearranged or if they hold any geometric significance.

3. **Rearranging the Reason**:
If we rearrange the equation, we can express it as:
\[ AE^2 + BF^2 + CD^2 = AF^2 + BD^2 + CE^2 \]
This implies a balance between these segments. However, this does not hold true for all triangles, as the segments are not necessarily equal.

**Hint**: Consider whether the segments can be equal in all cases or if there are specific conditions required.

4. **Conclusion on the Reason**:
Since the equality in the reason does not hold for all triangles, we conclude that the reason is incorrect.

**Hint**: Think about specific examples of triangles to test the validity of the reason.

5. **Final Evaluation**:
The assertion is correct while the reason is incorrect. Therefore, we conclude that:
- Assertion (A) is true.
- Reason (R) is false.

### Final Answer:
- Assertion (A) is correct.
- Reason (R) is incorrect.