Back to Math 445 Home Page | Back to Assignment Page | Next Assignment

Reading for Week 5

Lab Assignment E. Due Friday, 2/5/99

This assignment consists of the Portfolio Figures assigned in Lab #5.

Math 445. Assignment 5. Due Mon 2/8/99

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

• 5.1 (10 points) - (Angle defect greater than 0 implies multiple parallels) Sved, Chapter 4, page 77, #5
• 5.2 (10 points) - (Saccheri quadrilaterals) Sved, Chapter 4, page 78, #6

• 5.3 (10 points) - (DWEG circles are Apollonian) In Lab 4, recall that a DWEG circle with center A through B was obtained by inverting B through the circles representing DWEG lines to get points B' and B''. Then the DWEG circle was constructed as the circle through B, B' and B''. Prove that this circle is actually the Apollonian circle of A and O through B.

• 5.4. (10 points) - (Points symmetric by two mirrors) Given two disjoint circles c and d, prove that there is exactly one pair of points A and B such that the inversion of A is B, both with mirror c and with mirror d. Hint: If the circles are concentric, this is much easier.

• 5.5 (5 points) (Algebra and radical axis) Given two points A = (a, b) and C = (c,d) and a circle of radius r with center A and a circle of radius s with center C, find the equation of the radical axis in terms of a, b, c, d, r, s. Hint: Write down the definition and express the quantities algebraically in terms of the coordinates. Note: This formula is a simple proof that the radical axis is a line.

• 5.6 (5 points) (Algebra and Apollonian circle) Given two points A = (a, b) and C = (c,d) and a positive constant k, find the equation of the Apollonian circle of A and C with constant k in terms of a, b, c, d, k. Hint: Write down the definition and express the quantities algebraically in terms of the coordinates. Note: This formula is a simple proof that an Apollonian circle is a circle.

Do these problems and turn them in individually.

• 5.7. (10 points) - (P-circles are Apollonian) The P-circles were obtained by a construction parallel to the DWEG circles. Prove that a P-circle is also an Apollonian circle of points P and Q. (What are the points P and Q?)

• 5.8. (10 points) - (P-lines of symmetry)

  (a) Given two P-points A and B. Prove that there is exactly one P-line m which P-reflects A to B. (P-reflection is just inversion if the P-line is an arc and reflection if the P-line is a segment.)

  (b) Carry out a construction of m in a (random) example of points A and B.

  Comment: This line of symmetry of AB is the P-perpendicular bisector of AB, but here it is defined by a symmetry and not by distance. A useful consequence of this is that any point A in the P-disk model can be reflected to the center point; this can simplify a figure.

• If m is a P-line and A is a P-point, why is the inversion B of A in the support circle of m also a P-point? (In other words, why is B inside the P-disk and not outside?
• Given two parallel segments AB and CD, with midpoints M and N. If the centers of similitude of the segments are E and I, explain why E and I divide M and N harmonically. State and explain a similar relationship for the centers of similitude of two circles.
• Explain what happens to the Thales figure in non-Euclidean geometry: given lines OA and OB, and points A' on OA and B' on OB with OA'/OA = OB'/PB, is the triangle OA'B' similar to OAB? Explain your answer.
• Study the problems and constructions in Assignments 4 and 5 and the Lab Assignments.
• Write the correct formula for the inversion of point P(x,y) in the circle with center (a,b) and radius r. Also be able to find the dilation of P with the same center and ratio k.

Back to Math 445 Home Page | Back to Assignment Page | Next Assignment