Math 445. Assignment 6

Due Wed 2/4 at the beginning of class

This assignment has two parts that should be turned in separately (for they will be graded separately). Be sure to staple the two parts into separate packets.

Also, the first part is an INDIVIDUAL ASSIGNMENT and the second part is a GROUP assignment.

Part 1 (Figure Portfolio) (20 points)

Reading assignment. Sved, Chapter 3.

• Print out the figures for DWEG from the Math 487 lab, 1/28.

Part 2.

Problems: IMPORTANT. Each set of these problems are to be turned in by a GROUP of THREE or FOUR persons. Everyone is responsible for understanding all the ideas, but you can divide up the writing as you wish.

• 6.1 - (Parallels and transveral angles - one way only) Sved, Chapter 4, page 76, #1.
• 6.2 - (Angle sum for triangles) Sved, Chapter 4, page 76, #2.
• 6.3 - (What do parallels have to do with angle sums?) Sved, Chapter 4, page 77, #3.
• 6.4 - (Angle defect > 0 implies multiple parallels) Sved, Chapter 4, page 77, #5
• 6.5 - (Saccheri quadrilaterals) Sved, Chapter 4, page 78, #6

Reading: Ogilvy, Chapters 1, 2, 3 (except for the section on cross-ratio), and the two examples from Chapter 4 which were done in class and illustrated on the blue handouts, Friday, 1/30.

• Ideas from the HW above.
• Definition of harmonic division - ABCD.
• Definition of an Apollonian circle of A and B (as a locus)
• Proof that an Apollonian circle is really a circle.
• Proof that an Apollonian circle of A and B inverts A to B.
• How to construct an Apollonian circle of A and B through a point P.
• Relation between Apollonian circles and orthogonal circles.
• Relation between harmonic division, inversion and othogonal circles.
• Images of Euclidean figures under inversion as in Ogilvy, Chapter 4.