A median of a triangle is a line segment from a vertex to the midpoint of the opposite s...

A median of a triangle is a line segment from a vertex to the midpoint of the opposite side (Figure 1.45). Prove that the three medians of any triangle are concurrent (i.e., they have a common point of intersection) at a point G that is twothirds of the distance from each vertex to the midpoint of the opposite side. [Hint: In Figure 1.46, show that the point that is two-thirds of the distance from A to P is given by is two-thirds of the distance from B to Q and two-thirds of the distance from C to R.] The point G in Figure 1.46 is called the centroid of the triangle.