Find the coordinates of the points of trisection of the line segment joining (4, -1) and (-2, -3).

Let A(4, -1) and B(-2, -3) be the points of trisection of P and Q so AP = PQ = QB
For P: m1:m2 = AP:PB = 1:2
(x1, y1) = (4, -1) and (x2, y2) = (-2, -3)
∴x = (m1x2 + m2x1)/(m1 + m2) ⇒ x = (1 × -2 + 2 × 4)/(1 + 2) = -2 + 8/3 = 6/3 = 2
∴y = (m1y2 + m2y1)/(m1 + m2) = (1 × -3 + 2 × -1)/(1 + 2) = -3 - 2/3 = -5/3
∴P = (2, -5/3)
For Q: m1:m2 = AQ:QB = 2:1
(x1, y1) = (4, -1) and (x2, y2) = (-2, -3)
∴x = (m1x2 + m2x1)/(m1 + m2) ⇒ x = (2 × -2 + 1 × 4)/(1 + 2) = -4 + 4/3 = 0/3 = 0
∴y = (m1y2 + m2y1)/(m1 + m2) = (2 × -3 + 1 × -1)/(1 + 2) = -6 - 1/3 = -7/3
∴Q = (0, -7/3)