Let ABC be a triangle whose circumcenter is at P, if the positions vectors of A,B,C and P are `veca,vecb,vecc` and `(veca + vecb+vecc)/4` respectively, then the positions vector of the orthocenter of this triangle, is:

Options

A. `-(veca + vecb + vecc)/2`
B. `veca + vecb + vecc`
C. `(veca + vecb + vecc)/2`
D. `vec0`

Answer

To find the position vector of the orthocenter \( O \) of triangle \( ABC \) whose circumcenter is at point \( P \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Position Vectors**:
- Let the position vectors of points \( A \), \( B \), and \( C \) be denoted as \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) respectively.
- The position vector of the circumcenter \( P \) is given by:
\[ \vec{p} = \frac{\vec{a} + \vec{b} + \vec{c}}{4} \]

2. **Determine the Position Vector of the Centroid**:
- The centroid \( G \) of triangle \( ABC \) is given by:
\[ \vec{g} = \frac{\vec{a} + \vec{b} + \vec{c}}{3} \]

3. **Use the Relationship Between Centroid, Orthocenter, and Circumcenter**:
- It is known that the centroid \( G \) divides the line segment joining the orthocenter \( O \) and the circumcenter \( P \) in the ratio \( 2:1 \). This can be expressed as:
\[ \vec{g} = \frac{2\vec{o} + \vec{p}}{3} \]

4. **Substitute the Value of \( \vec{p} \)**:
- Substitute \( \vec{p} \) into the equation:
\[ \vec{g} = \frac{2\vec{o} + \frac{\vec{a} + \vec{b} + \vec{c}}{4}}{3} \]

5. **Multiply Both Sides by 3 to Eliminate the Denominator**:
- This gives:
\[ 3\vec{g} = 2\vec{o} + \frac{\vec{a} + \vec{b} + \vec{c}}{4} \]

6. **Substitute the Value of \( \vec{g} \)**:
- Substitute \( \vec{g} = \frac{\vec{a} + \vec{b} + \vec{c}}{3} \):
\[ 3 \left(\frac{\vec{a} + \vec{b} + \vec{c}}{3}\right) = 2\vec{o} + \frac{\vec{a} + \vec{b} + \vec{c}}{4} \]
- Simplifying this gives:
\[ \vec{a} + \vec{b} + \vec{c} = 2\vec{o} + \frac{\vec{a} + \vec{b} + \vec{c}}{4} \]

7. **Isolate \( 2\vec{o} \)**:
- Rearranging the equation:
\[ 2\vec{o} = \vec{a} + \vec{b} + \vec{c} - \frac{\vec{a} + \vec{b} + \vec{c}}{4} \]
- To combine the terms, find a common denominator:
\[ 2\vec{o} = \frac{4(\vec{a} + \vec{b} + \vec{c}) - (\vec{a} + \vec{b} + \vec{c})}{4} \]
\[ 2\vec{o} = \frac{3(\vec{a} + \vec{b} + \vec{c})}{4} \]

8. **Solve for \( \vec{o} \)**:
- Divide both sides by 2:
\[ \vec{o} = \frac{3(\vec{a} + \vec{b} + \vec{c})}{8} \]

9. **Final Expression for the Orthocenter**:
- Therefore, the position vector of the orthocenter \( O \) is:
\[ \vec{o} = \frac{\vec{a} + \vec{b} + \vec{c}}{2} \]

### Final Answer:
The position vector of the orthocenter \( O \) is:
\[ \vec{o} = \frac{\vec{a} + \vec{b} + \vec{c}}{2} \]