Math 302, Assignment 12, Fall 2000

Due Fri Dec 8 (at the beginning of class)

  1. (This is carried over from last week.)
    1. Let l be a line and let C be a circle with center P. Let m be the line through P that is perpendicular to l. Prove that if l and C intersect at two points A and B, then m is the perpendicular bisector of the segment AB.

      Hint: this is similar to the proof that if two circles intersect at two points A and B, then the line through the center of the circles is the perpendicular bisector of the segment AB.

    2. Consider a a vertical line and a circles in E^2 with center on the x axis. Intersect them with the upper half plane to get a vertical half line and a semi-circle. Prove that these two objects intersect in at most one point.

      Hint: use part one.

  2. Compute the area of each of the following three triangles. If a sphere were entirely covered with triangles of each of the following types, how many would fit? Show your calculations.
    1. A regular triangle with interior angle measure 120 degrees.
    2. A regular triangle with interior angle measure 90 degrees.
    3. A regular triangle with interior angle measure 72 degrees.
    Explain the relationship between this and the regular polyhedra whose faces are triangles.

  3. Prove that Euclid's Fifth Postulate (EFP) implies Playfield's Parallel Postulate (PPP) t on a space with absolute geometry.

    What part of your proof fails on S^2?

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For monday, read page 117-124.

New journal for Monday. We will be considering three new propositions.

  1. PT!, which appears on the top of 121.
  2. EFP, which appears on the top of 123.
  3. PPP, which appears on the top of 124.
For each of these, decide if it is true on E^2, on H^2, and on S^2. (Use the upper half plane model for H^2.) You can do 10.1 if you prefer, but 10.1 is harder.

The journal for Wed will be under ``handouts''.