Math 445 Assignment 5.

(Due Wed. 1/28 at the beginning of class)

Read Sved, Chapter 2.

1. (5 points) Sved, Chapter 2, page 43, #1 (tangent kites).
   Do a careful construction and turn it in, as well as a clear explanation. Note: This and other constructions can be found in Ogilvy.
2. (5 points) Sved, Chapter 2, page 43, #2. Do a careful construction of an example and turn it in. Write a clear explanation. This should be quite short if you use the theorems in the chapter.
3. (10 points) Sved, Chapter 2, page 43, #4. Do a careful construction of each of the 4 examples and turn them in (you probably want to use Sketchpad).
   Note: You can do them all by just computing a lot of inverses, but you should observe that if you use the theorems about inversion images, that you can do rather simple constructions of lines, circles, etc., that will give the construction without using the inversion construction directly.
4. (10 points) Sved, Chapter 2, page 43, #5. Do this as two parts.
   • (a) Given a segment AB on a tangent line to a circle c with center O and radius r, the image of the segment AB under inversion in c is an arc. Find the length of the arc in terms of the givens (this will involve angle AOB).
   • (b) Apply this to answer problem #5 in Sved.
5. (5 points) Center of an inverted circle.
   • (a) Let m be a circle. Construct with Sketchpad several examples, c1, c2, c3, c4, of circles and their inverses. Include in the figure the centers of the circles O1, O2, O3, O4, and their inverses O1', O2', O3', O4'. Observe whether the centers ever equal their inverses when you move the circles around. (Read the answer to Sved, Chapter 2, page 44, #7 to find out when this happens.)
   • (b) Construct a figure with a circle m and a circle c which are orthogonal. Let C be the center of c. Construct C' the inversion of C in m. Explain where C' is located in a simple way in terms of the given figure (i. e., what is a simple construction of C'?)
6. (10 points) Sved, Chapter 2, page 44, #8 (plus a figure).
   • (a) Construct with Sketchpad two circles c1 and c2 which do not intersect. Then construct two circles d1 and d2 which are each orthogonal to both c1 and c2. The circles d1 and d2 will intersect in two points A and B. Let m the the circle with center A which passes through B. Construct the circles (or lines) which are the inversions c1', c2', d1', and d2' of c1, c2, d1, and d2 in m.
   • Print out an example of this figure when c1 is interior to c2 and print another example which each of the circles is exterior to the other (you should be able to use the same Sketchpad sketch, just drag the circles).
   • (b) Explain briefly and clearly what are the objects in the figure c1', c2', d1', and d2' and how they are related. Explain why. (This is problem 8.)
7. (10 points) Composing inversions.
   Let m1 be a circle with center O and radius r1 and let m2 be a circle with the same center O and radius r2. Let J1 be inversion in m1 and let J2 be inversion in m2. (This means that J1 is the mapping or transformation so that for any point P, J1(P) = P', the inversion of P in m1.)
   • (a) If P is a point in the plane, and if Q = J2(J1(P)), how is |OQ| related to |OP|. (The answer should involve |OP| and the radii.)
   • (b) Find a dilation D (i.e., tell what is its center and ratio) so that the transformation D is exactly the same as J2J1. ("J2J1" means "J2 circle J1", the composition)
     Note: The proof of the theorem on pp. 34-35 of Sved uses this implicitly.
   • (c) Given J1 as above and a dilation E with center O and dilation ratio k, show that the composition EJ1 is an inversion (i.e., what is the center, what is the radius of the circle.
   • (d) In the (x,y) plane, let O = (0,0) and P = (x,y). Write down the formula in coordinates for J1, J2 and also J2J1 and check that this is consistent with (a) and (b).

Study questions for Quiz 3

• Study the HW problems, Assignment 5.
• Know the definition of inversion.
• Know an efficient and accurate staightedge-and-compass construction of the inversion of a point in a circle (wherever the point may be).
• Be able to use similar triangles to prove that the various constructions of inversion in Ogilvy, GTC, and Sved really construct the inverse.
• Again, be able to construct a circle through a certain set of points and orthogonal to a certain set of circles (e.g., through A and orthogonal to c1 and c2, or through A and B orthogonal to c1) using the concept of inversion.
• State and prove that the image of a line under inversion is what it is.
• State and prove that the image of a circle under inversion is what it is.
• Take a figure made up of lines and circles and invert in a circle whose center is some point in the figure. Explain qualitatively what the new figure looks like (make a fairly accurate sketch), or if necessary, be able to construct the inverted figure exactly. (Examples are in GTC chapter 10 from lab for 1/21. also from Sved and class.)
• Understand the conformal property of inversion and be able to use this to figure out what inverted figures look like, including especially an inversion of a pencil of circles and the pencil of circles orthogonal to the given pencil.
• (From class this week and Ogilvy.) Know the definition of harmonic division (a harmonic 4-sequence) and the circle of Apollonius.