If C is the middle point of AB and P is any point outside AB, then

Options

A. PA+PB=PC
B. PA+PB=2PC
C. PA+PB+PC=0
D. PA+PB+2PC=0

Explanation

To solve the problem, we need to prove that if C is the midpoint of AB and P is any point outside AB, then \( \vec{PA} + \vec{PB} = 2\vec{PC} \).

### Step-by-Step Solution:

1. **Define Points and Vectors**:
Let the position vectors of points A, B, C, and P be represented as:
- \( \vec{A} = \vec{0} \) (position vector of A)
- \( \vec{B} = \vec{b} \) (position vector of B)
- Since C is the midpoint of AB, we have:
\[ \vec{C} = \frac{\vec{A} + \vec{B}}{2} = \frac{\vec{0} + \vec{b}}{2} = \frac{\vec{b}}{2} \]
- Let the position vector of point P be \( \vec{P} \).

2. **Calculate Vectors PA and PB**:
The vector from P to A is given by:
\[ \vec{PA} = \vec{A} - \vec{P} = \vec{0} - \vec{P} = -\vec{P} \]
The vector from P to B is given by:
\[ \vec{PB} = \vec{B} - \vec{P} = \vec{b} - \vec{P} \]

3. **Add Vectors PA and PB**:
Now, we add \( \vec{PA} \) and \( \vec{PB} \):
\[ \vec{PA} + \vec{PB} = -\vec{P} + (\vec{b} - \vec{P}) = \vec{b} - 2\vec{P} \]

4. **Calculate Vector PC**:
The vector from P to C is given by:
\[ \vec{PC} = \vec{C} - \vec{P} = \frac{\vec{b}}{2} - \vec{P} \]

5. **Calculate Twice Vector PC**:
Now, we calculate \( 2\vec{PC} \):
\[ 2\vec{PC} = 2\left(\frac{\vec{b}}{2} - \vec{P}\right) = \vec{b} - 2\vec{P} \]

6. **Equate the Two Results**:
From steps 3 and 5, we have:
\[ \vec{PA} + \vec{PB} = \vec{b} - 2\vec{P} \]
\[ 2\vec{PC} = \vec{b} - 2\vec{P} \]

Therefore, we conclude that:
\[ \vec{PA} + \vec{PB} = 2\vec{PC} \]

### Conclusion:
We have proved that \( \vec{PA} + \vec{PB} = 2\vec{PC} \).