Let ABC to be an acute triangle with `BC=a,CA =b and AB=c, ` where `a ne b ne c.` From any point 'p' inside `Delta ABClet B,E,F` denot foot of perpendiculars form 'p' noto the sides, BC, CA and AB, respectively. Now, answer the following equations.
If `DeltaDEF` is equilateral, then 'P'

Options

A. coincides with incentre of `Delta ABC`
B. coincides with orthocentre of `Delta ABC`
C. lies on padal `Delta` of `Delta ABC`
D. None of the above

Answer

To solve the problem, we need to analyze the conditions under which triangle DEF, formed by the feet of the perpendiculars from point P to the sides of triangle ABC, is equilateral.

### Step-by-step Solution:

1. **Understanding the Triangle and Points**:
- Let triangle ABC be an acute triangle with sides BC = a, CA = b, and AB = c.
- Point P is inside triangle ABC, and we denote the feet of the perpendiculars from P to the sides BC, CA, and AB as points D, E, and F, respectively.

2. **Condition for Triangle DEF to be Equilateral**:
- For triangle DEF to be equilateral, the lengths of the sides DE, EF, and DF must be equal. This means:
\[ DE = EF = DF \]

3. **Using the Sine Rule**:
- The lengths of the segments can be expressed using the sine of the angles at point P:
\[ DE = PD \cdot \sin(\angle B), \quad EF = PE \cdot \sin(\angle C), \quad DF = PF \cdot \sin(\angle A) \]

4. **Setting the Equations**:
- Since DE = EF = DF, we can set up the following equations:
\[ PD \cdot \sin(\angle B) = PE \cdot \sin(\angle C) = PF \cdot \sin(\angle A) \]

5. **Finding Ratios**:
- From the above equations, we can derive ratios:
\[ \frac{PD}{PE} = \frac{\sin(\angle C)}{\sin(\angle B)}, \quad \frac{PE}{PF} = \frac{\sin(\angle A)}{\sin(\angle C)}, \quad \frac{PF}{PD} = \frac{\sin(\angle B)}{\sin(\angle A)} \]

6. **Conclusion about Point P**:
- The point P must satisfy these ratios, which implies that P lies on a specific locus defined by the angles of triangle ABC.
- When triangle DEF is equilateral, point P coincides with the intersection of the arcs of three circles defined by these ratios.

7. **Final Answer**:
- Therefore, if triangle DEF is equilateral, point P must lie at a specific location that does not coincide with the orthocenter of triangle ABC, nor does it lie on any parallel triangle. Thus, the correct option is:
- **None of the above**.