ABC is an equilateral triangle where AB = a and P is any point in its plane such that PA = PB + PC. Then `(PA^(2)+PB^(2)+PC^(2))/(a^(2))` is

Options

A. 3
B. `(sqrt(3))/(4)`
C. `(3)/(4)`
D. 2

Explanation

To solve the problem, we will follow a step-by-step approach to derive the required expression.

### Step 1: Understand the given information
We have an equilateral triangle ABC with sides AB = a. A point P exists such that PA = PB + PC. We need to find the value of the expression \( \frac{PA^2 + PB^2 + PC^2}{a^2} \).

### Step 2: Use the properties of the equilateral triangle
In an equilateral triangle, all angles are equal to 60 degrees. Therefore, if we consider the circumcircle of triangle ABC, point P lies on the arc BC. This means that the angle subtended at point P by points B and C is 120 degrees (since the angle subtended at the center is double that at the circumference).

### Step 3: Apply the Cosine Rule in triangle BPC
Using the cosine rule in triangle BPC:
\[ PB^2 + PC^2 - 2 \cdot PB \cdot PC \cdot \cos(120^\circ) = BC^2 \]
Since \( \cos(120^\circ) = -\frac{1}{2} \) and \( BC = a \), we can rewrite this as:
\[ PB^2 + PC^2 + PB \cdot PC = a^2 \]

### Step 4: Rearranging the equation
Rearranging the equation gives us:
\[ PB^2 + PC^2 = a^2 - PB \cdot PC \]

### Step 5: Substitute PA = PB + PC
We know from the problem statement that \( PA = PB + PC \). Squaring both sides gives:
\[ PA^2 = (PB + PC)^2 = PB^2 + PC^2 + 2PB \cdot PC \]

### Step 6: Substitute the expression for PB² + PC²
Now substituting the expression for \( PB^2 + PC^2 \) from Step 4 into the equation from Step 5:
\[ PA^2 = (a^2 - PB \cdot PC) + 2PB \cdot PC \]
\[ PA^2 = a^2 + PB \cdot PC \]

### Step 7: Find \( PA^2 + PB^2 + PC^2 \)
Now we can find \( PA^2 + PB^2 + PC^2 \):
\[ PA^2 + PB^2 + PC^2 = (a^2 + PB \cdot PC) + (a^2 - PB \cdot PC) = 2a^2 \]

### Step 8: Calculate the final expression
Now, we substitute this into our required expression:
\[ \frac{PA^2 + PB^2 + PC^2}{a^2} = \frac{2a^2}{a^2} = 2 \]

### Final Answer
Thus, the value of \( \frac{PA^2 + PB^2 + PC^2}{a^2} \) is **2**.