Math 445 Assignment 4A (Due Wed. 1/29)

Problem 4-1

Start with a triangle ABC and points A' on BC, B' on CA, C' on AB, so that AA', BB' and CC' are concurrent at D.  The barycentric coordinates of D are (x, y, z). Then construct point A'' by reflecting A' in the midpoint of BC (i.e., rotate by 180 degrees).  Similarly construct B'' by reflecting B' in the midpoint of CA and C'' by reflecting in the midpoint of AB.

• Are the lines AA'', BB'', CC'' concurrent at a point E?
• If so, what are the barycentric coordinates of point E?

Problem 4-2

In the (x, y) plane, A = (0,0), B = (1,0), C= (0,1).  Line m is the line with equation (x/a) + (y/b) = 1.

• Find the points of intersection A', B', C' of line m with lines BC, CA, AB.
• Find the 3 ratios AC'/C'B, BA'/A'C, CB'/B'A as for Ceva.
• Compute the product of these ratios.  Is it the same for any line m?
• How does this compare with what you get with Ceva?
• Will the product of these ratios be the same for any triangle ABC and any line m?  Explain.