p : If a triangle is equilateral then its centroid, circumcenter and incentre lies at same point
Converse of statement p is

Options

A. If centroid, circumcentre and incentre of a triangle lies at.same point then triangle is equilateral
B. If a triangle is not equilateral then its centroid, circumcenter and, incentre will not lie at same point
C. If a triangle is equilaterial then its centroid circumcenter and incentre will not lie at same point
D. If centroid, circumcenter .and incentre of a triangle does not lie at same point then it is not equilateral·

Answer

To find the converse of the statement \( p \): "If a triangle is equilateral, then its centroid, circumcenter, and incenter lie at the same point," we can follow these steps:

### Step 1: Identify the original statement
The original statement \( p \) can be expressed in the form of an implication:
- Let \( X \): "A triangle is equilateral."
- Let \( Y \): "The centroid, circumcenter, and incenter lie at the same point."

Thus, we can write the statement as:
\[ p: X \implies Y \]

### Step 2: Write the converse of the statement
The converse of an implication \( X \implies Y \) is given by \( Y \implies X \). This means we need to switch the hypothesis and conclusion.

### Step 3: Formulate the converse
From our definitions:
- The converse will be: "If the centroid, circumcenter, and incenter lie at the same point, then the triangle is equilateral."

We can express this formally as:
\[ \text{Converse of } p: Y \implies X \]
Which translates to:
\[ \text{If the centroid, circumcenter, and incenter lie at the same point, then the triangle is equilateral.} \]

### Step 4: Conclusion
Thus, the converse of the statement \( p \) is:
"If the centroid, circumcenter, and incenter lie at the same point, then the triangle is equilateral."