A(-4, 4), B(x, -1) and C(6, y) are the vertices of △ABC. If the centroid of this triangle ABC is at the origin, find the values of x and y.

A(2, 5), B(-1, 2) and C(5, 8) are the vertices of △ABC. P and Q are points on AB and AC respectively such that AP : PB = AQ : QC = 1 : 2.

(a) Find the coordinates of points P and Q.

(b) Show that BC = 3 × PQ.

A(-4, 2), B(0, 2) and C(-2, -4) are vertices of a triangle ABC. P, Q and R are mid-points of sides BC, CA and AB, respectively. Show that the centroid of △PQR is the same as the centroid of △ABC.

Given points A(1, 5), B(-3, 7) and C(15, 9).

(i) Find the equation of a line passing through the mid-point of AC and the point B.

(ii) Find the equation of the line through C and parallel to AB.

(iii) The lines obtained in part (i) and (ii) above, intersect each other at a point P. Find the coordinates of the point P.

(iv) Assign, giving reason, a special name of the figure PABC.