To solve the problem step by step, we will analyze both statements and derive the necessary coordinates and relationships.
### Step 1: Identify the Circumcenter and Centroid
The circumcenter of the triangle is given to be at the origin, which means its coordinates are:
\[
O(0, 0)
\]
The centroid is defined as the midpoint of the line segment joining the points (2, 3) and (4, 7). We can calculate the coordinates of the centroid \( G \) using the midpoint formula:
\[
G\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = G\left(\frac{2 + 4}{2}, \frac{3 + 7}{2}\right)
\]
Calculating this gives:
\[
G\left(\frac{6}{2}, \frac{10}{2}\right) = G(3, 5)
\]
### Step 2: Establish the Relationship Between Circumcenter, Centroid, and Orthocenter
According to the problem, the circumcenter \( O \), centroid \( G \), and orthocenter \( H \) are collinear, and the centroid divides the segment \( OH \) in the ratio \( 1:2 \).
Let the coordinates of the orthocenter \( H \) be \( (x_1, y_1) \). Since \( G \) divides \( OH \) in the ratio \( 1:2 \), we can use the section formula to find \( H \):
\[
G = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
where \( O(0, 0) \) is \( (x_2, y_2) \) and \( H(x_1, y_1) \) is \( (x_1, y_1) \), with \( m = 1 \) and \( n = 2 \).
Setting up the equations:
\[
3 = \frac{1 \cdot 0 + 2 \cdot x_1}{1 + 2} \quad \text{and} \quad 5 = \frac{1 \cdot 0 + 2 \cdot y_1}{1 + 2}
\]
### Step 3: Solve for \( x_1 \) and \( y_1 \)
From the first equation:
\[
3 = \frac{2x_1}{3} \implies 2x_1 = 9 \implies x_1 = \frac{9}{2} = 4.5
\]
From the second equation:
\[
5 = \frac{2y_1}{3} \implies 2y_1 = 15 \implies y_1 = \frac{15}{2} = 7.5
\]
Thus, the coordinates of the orthocenter \( H \) are:
\[
H\left(4.5, 7.5\right)
\]
### Step 4: Verify the Condition \( 5x - 3y = 0 \)
We need to check if the coordinates of the orthocenter satisfy the equation \( 5x - 3y = 0 \):
Substituting \( x = 4.5 \) and \( y = 7.5 \):
\[
5(4.5) - 3(7.5) = 22.5 - 22.5 = 0
\]
This confirms that the orthocenter satisfies the equation.
### Conclusion
Both statements are true:
- Statement 1 is true because the orthocenter satisfies the relation \( 5x - 3y = 0 \).
- Statement 2 is true as the circumcenter, centroid, and orthocenter are collinear and the centroid divides the segment in the ratio \( 1:2 \).
