Math 403: Homework 3
Book problems: 1.24, 2.1
In problem 1.24 a transversal of triangle ABD is just a line intersecting all three sides of the triangle (if the sides are extended far enough).
In problem 2.1, if you want to use the argument in the back of the text, then you should prove the statements that are made there.
P1. Suppose that α is a bijection. Show that the inverse of α is unique. That is, if βα = δα, then β=δ.
P2. Let A1, B1, C1 lie on a line, and let A2, B2, C2 lie on a second line. Suppose that the line lA1B2 meets the line lB1A2 at a point D, that the line lA1C2 meets the line lC1A2 at a point E, and that the line lB1C2 meets the line lC1B2 at a point F. Finally, suppose that the three lines lC1A2, lB1C2, and lA1B2 are not parallel and therefore describe a triangle. Use Menelaus' theorem to show that the points D, E, and F are collinear.
Hint: If the corners of the triangle are called U, V, and W, then apply the theorem to this triangle with the five following choices of opposite side points: (1) A1, E, C2, (2) C1, F, B2, (3) B1, D, A2, (4) A1, B1, C1, and (5) A2, B2, C2.
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