Problem Set 5 Part 2

Suppose a point P has barycentric coordinates a, b, c with respect to triangle ABC. If line AP intersects line BC in A', line BP intersects line CA in B' and line CP intersects line AB in C'. compute the ratios on the sides: AC'/C'B, BA'/A'C, CB'/B'A in terms of a, b, c. Hint: draw the parallels to the sides through P as in class and find the lengths.

Draw a quadrilateral ABCD and the midpoint triangle. Show that the sides of the midpoint quadrilateral are parallel using vectors.

In a parallelogram ABCD, show that the center of mass is the point of intersection of the diagonals.

Do these problems.

1. Given points A(1,3) and B(23, 171), what is the midpoint of AB? What is the equation of the perpendicular bisector of AB?
2. If mass 1 is placed at each of the points below, what is the center of mass M? What special lines in triangle ABC are the lines AM, BM, CM?

A(2,3), B(25, 62), C(101, 3)

3. Where does the line through A(10,15) and B (6, 9) intersect the line with equation 2x + 3y = 12?
4. In 3-space, given a mass 1 at A(1, 2, 1) , mass 2 at B(-1,0,2) and mass 1 at C(4, -2, 1), what is the barycenter M (center of mass) of this system?