(a) Let the vertices of ∆ABC be A(−5, 3), B(7, 7), and C(3, 1). Find the point on each med...

(a) Let the vertices of ∆ABC be A(−5, 3), B(7, 7), and C(3, 1). Find the point on each median that is two-thirds of the way from the vertex to the midpoint of the opposite side. (Recall that a median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side.) What do you observe? Hint: Use the result in Exercise.

(b) Follow part (a) but take the vertices to be A(0, 0), B(2a, 0), and C(2b, 2c). What do you observe? What does this prove?

Exercise

Let P1(x1, y1) and P2(x2, y2) be two given points. Let Q be the point .

(a) Show that the points P1, Q, and P2 are collinear. Hint: Compute the slope of and the slope of .

(b) Show that . (In other words, Q is on the line segment and two-thirds of the way from P1 to P2.)