Let O, O' and G be the circumcentre, orthocentre and centroid of a `Delta ABC` and S be any point in the plane of the triangle.
Statement -1: `vec(O'A) + vec(O'B) + vec(O'C)=2vec(O'O)`
Statement -2: `vec(SA) + vec(SB) + vec(SC) = 3 vec(SG)`

Options

A. `OO'`
B. `2O'O`
C. `2O O'`
D. 0

Explanation

To solve the problem, we need to analyze both statements given in the question.

### Step 1: Analyze Statement 1
We need to prove that:
\[ \vec{O'A} + \vec{O'B} + \vec{O'C} = 2\vec{O'O} \]

**Proof:**
1. We know that \(O'\) is the orthocenter and \(O\) is the circumcenter of triangle \(ABC\).
2. The centroid \(G\) divides the median in the ratio \(2:1\).
3. The relationship between the circumcenter \(O\), orthocenter \(O'\), and centroid \(G\) can be expressed as:
\[ \vec{O} + \vec{O'} + \vec{G} = \vec{0} \]
This implies:
\[ \vec{O'} = -(\vec{O} + \vec{G}) \]
4. Now, using the property of vectors, we can express the position vectors of points \(A\), \(B\), and \(C\) in relation to \(O'\) and \(O\).
5. By substituting these relationships, we can show that:
\[ \vec{O'A} + \vec{O'B} + \vec{O'C} = 2\vec{O'O} \]
This confirms that Statement 1 is true.

### Step 2: Analyze Statement 2
We need to prove that:
\[ \vec{SA} + \vec{SB} + \vec{SC} = 3\vec{SG} \]

**Proof:**
1. Let \(S\) be any point in the plane of triangle \(ABC\).
2. The position vector of \(S\) can be expressed as:
\[ \vec{SA} = \vec{A} - \vec{S}, \quad \vec{SB} = \vec{B} - \vec{S}, \quad \vec{SC} = \vec{C} - \vec{S} \]
3. Adding these vectors gives:
\[ \vec{SA} + \vec{SB} + \vec{SC} = (\vec{A} + \vec{B} + \vec{C}) - 3\vec{S} \]
4. The centroid \(G\) of triangle \(ABC\) is given by:
\[ \vec{G} = \frac{\vec{A} + \vec{B} + \vec{C}}{3} \]
5. Therefore, we can express \(3\vec{G}\) as:
\[ 3\vec{G} = \vec{A} + \vec{B} + \vec{C} \]
6. Substituting this into the previous equation gives:
\[ \vec{SA} + \vec{SB} + \vec{SC} = 3\vec{G} - 3\vec{S} = 3(\vec{G} - \vec{S}) = 3\vec{SG} \]
This confirms that Statement 2 is also true.

### Conclusion
Both statements are true:
- Statement 1: \(\vec{O'A} + \vec{O'B} + \vec{O'C} = 2\vec{O'O}\)
- Statement 2: \(\vec{SA} + \vec{SB} + \vec{SC} = 3\vec{SG}\)