King ...

To find the intersection points of the given lines, we need to solve the system of equations formed by the three lines.

x - 3y = 1 ...(1)

x + y + 1 = 0 ...(2)

2x - 3y + 4 = 0 ...(3)

Solving equations (1) and (2), we get:

x - 3y = 1 ...(1)

x + y = -1 ...(4)

Adding equations (1) and (3), we get:

3x + 3 = 0

x = -1

Substituting x = -1 in equation (4), we get:

-1 + y = -1

y = 0

Therefore, the intersection point of lines (1) and (2) is (-1, 0).

Substituting x = -1 in equation (3), we get:

-2 - 3y + 4 = 0

y = -2/3

Substituting y = -2/3 in equation (1), we get:

x - 3(-2/3) = 1

x = -1/3

Therefore, the intersection point of lines (1) and (3) is (-1/3, -2/3).

Substituting y = -2/3 in equation (2), we get:

x + (-2/3) + 1 = 0

x = 1/3

Therefore, the intersection point of lines (2) and (3) is (1/3, -2/3).



To find the midpoint of the side formed by the intersection points (-1, 0) and (-1/3, -2/3), we use the midpoint formula:

((x1 + x2)/2, (y1 + y2)/2)

Substituting (-1, 0) and (-1/3, -2/3), we get:

((-1 + (-1/3))/2, (0 + (-2/3))/2)

(-2/3, -1/3)

Therefore, the midpoint of the side formed by the intersection points (-1, 0) and (-1/3, -2/3) is (-2/3, -1/3).

Similarly, we can find the midpoints of the other two sides of the triangle:

Midpoint of the side formed by (-1, 0) and (1/3, -2/3):

(((-1) + (1/3))/2, ((0) + (-2/3))/2)

(-2/3, -1/3)

Midpoint of the side formed by (-1/3, -2/3) and (1/3, -2/3):

(((-1/3) + (1/3))/2, ((-2/3) + (-2/3))/2)

(0, -4/3)