Find the center of gravity of the triangle ABC, if a (-4; 10), in (2; 8), with (-4; -6).

1. The center of gravity of the triangle is at the intersection of the medians.

2. Find the coordinates of the point M – the midpoint of the side BC:

A (-4; 10);
B (2; 8);
C (-4; -6).
x (M) = (2 – 4) / 2 = -2/2 = -1;
y (M) = (8 – 6) / 2 = 2/2 = 1;
M (-1; 1).
3. The point O of the intersection of the medians divides them in a ratio of 2: 1:

AO: MO = 2: 1.

Therefore, the coordinates of point O can be determined by the formula:

x (O) = x (M) + (x (A) – x (M)) / 3;
y (O) = y (M) + (y (A) – y (M)) / 3;
x (O) = -1 + (-4 + 1) / 3 = -1 + (-3) / 3 = -1 – 1 = -2;
y (O) = 1 + (10 – 1) / 3 = 1 + 9/3 = 1 + 3 = 4;
O (-2; 4).
Answer: (-2; 4).