Reading: Read Berele-Goldman 10.1 and 10.2. Also there will be web references.
Theorem:
Given any triangle ABC and a point P on line AB. Construct the points P1,m …, P5 as follows:
Then, the line P5P is parallel to BC, so that the figure PP1P2P3P4P5 is a (self-intersecting) hexagon with each side parallel to one of the sides of ABC.
Prove the theorem by this method:
Method 1. Prove this theorem for the special case where the triangle in the theorem is an equilateral triangle EFG. (The figure has lots of similar and congruent figures in this case.) Then, since any triangle ABC is the image of EFG by an affine transformation, the theorem is true for any ABC. (This is true since the relationships in the theorem are affine invariant.)
Prove the theorem by this method:
Method 2. Let the barycentric coordinates of P with respect to triangle ABC be (u, v, 0), i.e., P = uA + vB + 0C, where u+v=1.
Write down the barycentric coordinates of the points P1, …, P5. Then check from the barycentric coordinates of P5 that PP5 is parallel to BC.
In Lab 1 Part 3 the quadratic Bezier curve was constructed. In this problem we use the same notation as in the lab.
(a) If the ratio t = ¾ and the points A1 = (2,3), A2 = (-1, 4), A3 = (-3,8), compute numerically the points B1, B2, and C1.
(b) If the ratio t = t (a variable t) and the points A1 = (2,3), A2 = (-1, 4), A3 = (-3,8), compute numerically the points B1, B2, and C1. (The points will depend on t.) For which points is the expression in t a polynomial of degree 1? For which ones a polynomial of degree 2?
(c) If the ratio is t and the points are A1, A2, A3 (no numerical values specified), write the formula for B1, B2, and C1.
(d) Write the formula for the cubic Bezier. Specifically, given points A1, A2, A3, A4, compute the point D1 as a formula in the points and t.
Given a triangle ABC and points A' on BC, B' on CA, C' on AB with BA'/BC = CB'/CA = AC'/AB = t, find the ratio (area A'B'C')/(area ABC) as a function of t.
Construct the incircle of the triangle ABC.