To solve the problem, we need to find the coordinates of points D, E, and F, which divide the sides BC, CA, and AB of triangle ABC in the ratio 2:1. Then we will find the centroid of triangle DEF and compare it with the centroid of triangle ABC.
### Step 1: Find the coordinates of point D
Point D divides line segment BC in the ratio 2:1.
- Coordinates of B: \( B(-1, 3) \)
- Coordinates of C: \( C(3, 4) \)
Using the section formula, the coordinates of point D can be calculated as follows:
\[
D\left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right)
\]
Where \( m = 2 \) and \( n = 1 \).
\[
D\left(\frac{2 \cdot 3 + 1 \cdot (-1)}{2 + 1}, \frac{2 \cdot 4 + 1 \cdot 3}{2 + 1}\right)
\]
Calculating the x-coordinate of D:
\[
x_D = \frac{6 - 1}{3} = \frac{5}{3}
\]
Calculating the y-coordinate of D:
\[
y_D = \frac{8 + 3}{3} = \frac{11}{3}
\]
So, the coordinates of D are:
\[
D\left(\frac{5}{3}, \frac{11}{3}\right)
\]
### Step 2: Find the coordinates of point E
Point E divides line segment CA in the ratio 2:1.
- Coordinates of C: \( C(3, 4) \)
- Coordinates of A: \( A(1, 2) \)
Using the section formula again:
\[
E\left(\frac{2 \cdot 1 + 1 \cdot 3}{2 + 1}, \frac{2 \cdot 2 + 1 \cdot 4}{2 + 1}\right)
\]
Calculating the x-coordinate of E:
\[
x_E = \frac{2 + 3}{3} = \frac{5}{3}
\]
Calculating the y-coordinate of E:
\[
y_E = \frac{4 + 4}{3} = \frac{8}{3}
\]
So, the coordinates of E are:
\[
E\left(\frac{5}{3}, \frac{8}{3}\right)
\]
### Step 3: Find the coordinates of point F
Point F divides line segment AB in the ratio 2:1.
- Coordinates of A: \( A(1, 2) \)
- Coordinates of B: \( B(-1, 3) \)
Using the section formula:
\[
F\left(\frac{2 \cdot (-1) + 1 \cdot 1}{2 + 1}, \frac{2 \cdot 3 + 1 \cdot 2}{2 + 1}\right)
\]
Calculating the x-coordinate of F:
\[
x_F = \frac{-2 + 1}{3} = \frac{-1}{3}
\]
Calculating the y-coordinate of F:
\[
y_F = \frac{6 + 2}{3} = \frac{8}{3}
\]
So, the coordinates of F are:
\[
F\left(\frac{-1}{3}, \frac{8}{3}\right)
\]
### Step 4: Find the centroid of triangle DEF
The centroid \( G \) of triangle DEF is given by:
\[
G\left(\frac{x_D + x_E + x_F}{3}, \frac{y_D + y_E + y_F}{3}\right)
\]
Substituting the coordinates of D, E, and F:
\[
G\left(\frac{\frac{5}{3} + \frac{5}{3} + \frac{-1}{3}}{3}, \frac{\frac{11}{3} + \frac{8}{3} + \frac{8}{3}}{3}\right)
\]
Calculating the x-coordinate of G:
\[
x_G = \frac{\frac{5 + 5 - 1}{3}}{3} = \frac{9}{9} = 1
\]
Calculating the y-coordinate of G:
\[
y_G = \frac{\frac{11 + 8 + 8}{3}}{3} = \frac{27}{9} = 3
\]
So, the centroid of triangle DEF is:
\[
G(1, 3)
\]
### Step 5: Find the centroid of triangle ABC
The centroid \( G_{ABC} \) of triangle ABC is given by:
\[
G_{ABC}\left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right)
\]
Substituting the coordinates of A, B, and C:
\[
G_{ABC}\left(\frac{1 + (-1) + 3}{3}, \frac{2 + 3 + 4}{3}\right)
\]
Calculating the x-coordinate of \( G_{ABC} \):
\[
x_{G_{ABC}} = \frac{3}{3} = 1
\]
Calculating the y-coordinate of \( G_{ABC} \):
\[
y_{G_{ABC}} = \frac{9}{3} = 3
\]
So, the centroid of triangle ABC is:
\[
G_{ABC}(1, 3)
\]
### Conclusion
Both centroids \( G \) and \( G_{ABC} \) are the same, which means:
- Statement I: The centroid of triangle DEF is (1, 3). **True**
- Statement II: The triangle ABC and DEF have the same centroid. **True**
