Math 445 Assignment 3, Due Wednesday, 1/14/98

1. Centers of similitude of two segments (20 points)
   • (a) Create a Sketchpad figure with two parallel segments AB and CD; let k = |CD|/|AB|. Construct the two centers of similitude of the two segments; let O be the center with positive ratio and O' be the center with negative ratio.
   • Construct the midpoints M of AB and N of CD. Draw the line MN and observe that the two centers of similitude appear to be on the line.
   • (b) Explain why these four points are collinear.
   • (c) Use similar triangles to derive the signed ratios ON/OM and O'N/O'M in terms of k.
   • (d) Construct the points P = (1-(-1/2))A + (-1/2)B and Q = (1-(-1/2))C + (-1/2)D. Draw the lines OP and O'P. which of these two lines appears to pass through Q? Explain why this line does in fact pass through Q.
   • (e) Let S be the intersection of the other of the two lines OP and O'P with line CD. Tell the number s such that S = (1- s)C + sD.
   • (f) Let D be the dilation with center O which takes Q to P. What is the ratio of similitude of this dilation? Let D' be the dilation with center O' which takes P to S. What is the ratio of similitude of this dilation? What is the ratio of similitude of the composition D'D? What is the center (i.e., the fixed point) of D'D?
   
   
2. Centers of similitude of two circles, starting with tangents (10 points)

   Review. Suppose m and n are two lines intersecting at a point P, forming an angle a (and corresponding vertical angle a'). Let c1 and c2 be two circles inside angle a or a' which are tangent to both m and n. Then the centers E1 and E2 of the circles lie on the bisector of angle a. We denote the following points: line m is tangent to the circles c1 and c2 at M1 and M2 and the line n is tangent to the circles at N1 and N2.

   • (a) Prove that these signed ratios are equal: PE2/PE1 = PM2/PM1 = PN2/PN1.
   • (b) Explain how to conclude from (a) that the dilation D with center P and ratio PE2/PE1 takes circle c1 to c2, and also takes E1 to E2, M1 to M2 and N1 to N2 .
   • (c) If G1 is a point on c1, explain why one of the points of intersection of line PG1 with circle c2 is G1', the image of G1 under the dilation with center P that takes c1 to c2.
   
   
3. Centers of similitude of two circles, starting with circles (10 points)

   Next, we start with two circles and nothing else, but then construct the tangent lines and the center of similitude.

   • (a) Create a Sketchpad figure with two circles AB and CD (by circle AB we mean the circle with center A through B. etc.) and construct the two centers of similitude. Let O be the center with positive ratio and O' be the center with negative ratio. Let k = |CD|/|AB|. Make and save a script "Circle sim centers" that takes AB and CD as givens and constructs O and O'.
   • (b) What are the signed ratios OC/OA and O'C/O'A in terms of k? Explain.
   • (c) Move the circles so that they are exterior to one another and construct the two tangent lines from O to c1 and the two tangent lines from O' to c1. Observe that these four lines appear to be the common tangents. Explain why this is true.
   
   
4. Similitude composition (Composition is a similitude--usually) (10 points)
   • Write down the vector formula for the dilation with center P and ratio k.
   • Write down the 3x3 matrix of this dilation, as in Sibley. What is the upper-left 2x2 matrix of the matrix of any dilation with ratio k?
   • Use the vector or matrix formula for similitudes to show that the composition U = TS of two similitudes S and T, possibly with different centers, is a similitude (with one special case which is an exception; what is it?). How is the ratio of similitude of U related to the ratios of S and T?
   
   
5. Applications of composition (10 points)

   (a) Theorem of Menelaus

   • Construct a Sketchpad figure that illustrates how the Theorem of Menelaus can be proved using composition of dilations.

   (b) Three Circle Theorem

   • Construct a Sketchpad figure with 3 circles and then use the script "Circle sim centers" to construct the centers of similitude of each pair.
   • Construct the lines through pairs of centers and observe that some centers are collinear.
   • Write down an explanation of this in terms of centers of similitude.
   • What are the special cases when one or more of the centers do not exist.