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Reading for Week 1 and 2

Read Sved Introduction and Chapter 1. Read and study chapter 1 carefully.

The concepts of radical axis and orthogonal circles are also discussed briefly in B&B and Bix.

Math 445 Assignment 2. Due Wed., 1/12/00 (55 points)

Note: The problems in Sved have answers in the back. This is why the number of points is less for these problems than what they otherwise might be. You are getting credit for taking the trouble to write up the answers in your own words, putting in any missing details. You will be asked to present problems in class.

Any of these questions or constructions can appear on the quizzes, so you should learn the facts and constructions and also master the relationships needed to prove these facts.

1. (5 points) Sved, Chapter 1, page 23, #1. This is review.
2. (5 points) Sved, Chapter 1, page 23, #2.
3. (5 points) Sved, Chapter 1, page 24, #4.
4. (5 points) Sved, Chapter 1, page 24, #5. Draw two circles that do not intersect and construct the radical axis using your quick method.
5. (5 points) Sved, Chapter 1, page 24, #9.
6. (10 points) You know the inscribed angle theorem from B&B (and the Carpenter's principle, which is a special case). On page 13 of Sved, there are three figures with equal angles marked. Show why each equal-angle relationship is true.
   (You may need an extension of the inscribed angle theorem to angles formed by tangent lines.)
7. (10 points) Power of a point with Coordinates.

• (a) Let c1 be the circle with center O1(0,0) and radius r1. What is the power of c1 at P(x,y)? (i.e., give a formula for the power in terms of x and y)
• (b) Let c2 be the circle with center O2(s,0) and radius r2. What is the power of c2 at P(x,y)? Note that this formula still is valid if r2=0, as in GTC, Chapter 9.
• (c) Find the equation in x and y for the radical axis of c1 and c2.
• (d) Use your answer to verify the formula in Sved, Chapter 1, page 23, #3.

1. (10 points) Construction challenge. Draw a line m and two points A and B on the same side of the line (A and B should be at random). Construct all circles c which pass through A and B and which are tangent to m. Describe the steps in your method. Hint. Sved figure on page 16 and the corresponding parts of Lab 1..

Study Questions that you should be able to answer (they may appear on a quiz).

• What is the definition of the radical axis of two circles? (Note: the word "line" does not appear in this.)
• How do we know that the radical axis is a line?
• If two circles intersect, why does the radical axis pass through the points of intersection?
• Suppose that two circles c1 and c2 have a common tangent ST, then show the radical axis of the two circles passes through the midpoint of ST.
• In GTC, the radical axis of a circle C1 and a point P is defined. Explain why this is the usual definition in terms of powers of a point if you allow a "circle" to have radius 0.
• Let m be the radical axis of circle c1 and point P. If O is on m, explain why circle d, with center O and orthogonal to c1, must pass through P.
• Know this construction: Given a circle c1 and a point P1, construct a circle centered at P1 which is orthogonal to c1. (Is there more than one circle? Does it always exist?)
• Know this construction: Given a circle c1 and a point P1, construct a circle through P1 which is orthogonal to c1. (Is there more than one circle? Does it always exist? Notice that this construction is quite different from the previous one.)
• Figure out this set of related constructions. Given p circles and q points, find a circle (or circles) through the q points which is (are) orthogonal to each of the p circles. This works if p+q = 3 or less.
• Given two circles c1 and c2 and a point P, when can you construct a circle with center P which is orthogonal to both the given circles?

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