If in a `DeltaABC, AD, BE and CF` are the altitudes and R is the circumradius, then the radius of the circumcircle of `DeltaDEF` is

Options

A. 2R
B. R
C. `R/2`
D. None of these

Answer

To find the radius of the circumcircle of triangle DEF formed by the feet of the altitudes from vertices A, B, and C of triangle ABC, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have triangle ABC with altitudes AD, BE, and CF. The points D, E, and F are the feet of these altitudes. We need to find the circumradius (R') of triangle DEF in terms of the circumradius (R) of triangle ABC.

**Hint**: Visualize triangle ABC and identify points D, E, and F.

2. **Using the Property of Circumradius**:
It is known that for any triangle, the circumradius of the triangle formed by the feet of the altitudes (triangle DEF) is related to the circumradius of the original triangle (triangle ABC). Specifically, the circumradius of triangle DEF is half of that of triangle ABC.

**Hint**: Recall the relationship between the circumradius of a triangle and the triangle formed by its altitudes.

3. **Applying the Formula**:
Given that the circumradius of triangle ABC is R, we can express the circumradius of triangle DEF as:
\[ R' = \frac{R}{2} \]

**Hint**: Remember that this relationship holds true for any triangle and its orthocenter.

4. **Conclusion**:
Therefore, the radius of the circumcircle of triangle DEF is \( \frac{R}{2} \).

**Hint**: Check if this result aligns with known properties of triangles and their circumradii.

### Final Answer:
The radius of the circumcircle of triangle DEF is \( \frac{R}{2} \).