Assignment 2B (Due Fri 1/21)

Reading: Read GTC Chapter 9.

Review section 5.5 in Berele-Goldman and also especially Problem 5.1 and Figure 5.22.

Problem 1

  1. In this figure, tell why the product |OC||OD| = |OE||OF| for any two circles through A and B.
  2. Draw such a picture on a sheet of paper and construct the circle d with center O that is orthogonal to one of the circles.
  3. Explain why the circle d must be orthogonal to both circles if it is orthogonal to one of the circles.
  4. Draw any line OP through O. Let M and N be the intersections of this line with circle d. Construct the circle m through A, B, M and the circle n throught A, B, N.
  5. Explain why the circles m and n are the circles through A and B that are tangent to line OP.

Problem 2 (Concurrence of Radical Axes)

Based on the definition of the power function and the radical axis of two circles, prove that if a, b, c are 3 circles and k, m, n are the radical axes of b and c, c and a, a and b, respectively, then the 3 lines k, m, n are either parallel or concurrent.

Hint: This proof is a lot like the proof of concurrence of perpendicular bisectors. Review that proof.

Problem 3 (Shortcuts to Constructing Radical Axis)

For this problem, use this theorem and your answer to Problem 2 to answer A-D below.

Theorem: The radical axis of two (non-concentric) circles is a line perpendicular to the line of centers of the circles. (Proof given in class.)

  1. Suppose c and d are two circles that intersect at two points. Using this theorem, explain why the radical axis of the two circles is the line through the two points of intersection.
  2. Suppose c and d are two circles that intersect at one point. Using this theorem, explain why the radical axis of the two circles is the common tangent line to the two circles.
  3. Suppose c and d are two circles that do not intersect. Explain how one can construct a point on the radical axis (and then the radical axis) by drawing a third circle e "at random" so that 3 intersects c and d in two points each.
    Hint: Problem 2.
  4. D. Draw two disjoint circles of unequal size and use the method of C to construct the radical axis in this example.

Problem 4 (Constructing an Orthogonal Circle)

Draw a circle c, a line m that does not intersect c and a point A inside the circle c. Construct a circle d through A that is orthogonal to both c and m.

Write a clear description of the major steps of your construction. (The outline should be at the the level of "construct the perpendicular bisector, invert the point, construct the tangent, construct the radical axis, etc, not a description of every arc.)

Explain briefly but convincingly why your construction works.