Quiz 2 Study

What won't be on the test

Spherical geometry and the DWEG model will not appear as such on the test. However, as you should have noticed, the same constructions involving circles keep coming up in different contexts, so the constructions will be on the test.

What may be on the test

Quiz 2 will start with the work on orthogonal circles that the class began Monday 2/3 and finish with Assignment 7 and the 445 class Wed 2/19.

Basic Definitions and Theorems

For this, reread the assigned reading, Assignment 7, class handouts, Lab 6.

• Sved - Chapters 1, 2 (including the problems)
• Ogilvy - Chapters 1, 2, 3 (through p. 39). Also, Apollonian circles will not appear again (yet).
• Geometry through the Circle - Chapter 9 (parts in lab + all constructions of orthogonal circles in 9.1). For circle construction strategy, you may wish to read 6.1, 6.2, 6.3.
• Definitions: Power of a point, radical axis, orhtogonal circles, inversion of a point in a circle, harmonic division, Apollonian circles (in Ogilvy).

Theorems: Center of a circle orthogonal to two circles must be on the radical axis. Circle through point P and its inverse P' must be orthogonal to the circle of inversion (i.e. the mirror circle). The inversion of a line is a circle (or a line -- which one- give details and know proof). The inversion of circle in a circle is a circle or a line (which one -- give details). Relation between harmonic division and inversion.

• Prove (using some other theorems): If P' is the inversion of P in circle c, any circle d through P and P' orthogonal to c.
• Prove that the inversion image of a line is either a circle or a line. Given a figure with a circle c, a line m and a circle d, construct the images by inversion in c: m' and d'.
• How is harmonic division related to inversion (state the relationship).

Constructions

In recent assignments and the reading we have learned these constructions:

Basic circle constructions

• Given a point A and a circle c, construct a circle d with center A orthogonal to c.
• Construct the radical axis of two circles
• Construct the nversion of point in a circle (inside or outside the circle of inversion)

Constructing Orthogonal Circles

Note: Some of these constructions have special cases when points are on circles, circles are in special positions, etc. Most of these constructions have appeared on assignments and tests going back to 444.

• Given THREE points and ZERO circles, construct a circle through the points and orthogonal to the circles.
• Given TWO points and ONE circles, construct a circle through the points and orthogonal to the circles.
• Given ONE points and TWO circles, construct a circle through the points and orthogonal to the circles.
• Given ZERO points and THREE circles, construct a circle through the points and orthogonal to the circles.

Constructing Inversion Images of Sets

Note: Again, there are special cases. Some cases are much simpler when the object intersects the inverting circle (because the points of intersection do not move).

• Construct the image under inversion of a line.
• Construct the image under inversion of a circle.

Constructing Harmonic Sets

• Given A, B, C on a line, construct D so that CD divides AB harmonically. (Know how to do it with just a straightedge.)

References on the web from 444-445 sites

Math 444, Au 2002

• Lab 10, Part 1 (Apollonian circles)
• Assignment 10B, problem 10-3 (exterior bisector thm)
• Assignment 10B, problem 10-5 (stereographic construction of inverse)

Math 445, W 2002

There is a lot of review material on the 445 Test and Study page link. You will have to sort out what is relevant from what is not, alas. Notice that the calendar was different last year, so the inversion questions are on Quiz 1, Quiz 2 and the Midterm.

There are other links to old versions of 445 from this page.