Math 445 Midterm: NAME _________________________________

Do all 4 problems. Note choice in problem 4.

Problem 1: Circle orthogonal to 3 circles

Construct a circle d that is orthogonal to all the three circles c1, c2, c3. (The points O1, O2, and O3 are the centers of the circles, provided for your convenience.) Label key elements clearly and make clear how key elements are constructed.

Problem 2: Apollonian Circles

Answer questions (a) - (f) on this and the following page. For true/false, just give answer, not explanation.

(a) Given two points A and B, state the definition of an Apollonian circle of A and B.

(b) Given points A, B and C below, construct the Apollonian circle of A and B that passes through C.

Problem 2 (continued)

(c) Is this True or False? If c is an Apollonian circle of A and B and d is a circle through A and B, then c and d are orthogonal.

(d) Suppose c is a circle in the Euclidean plane, and A is a point not on c. If A' is the inversion of A in c, is it true or false that c must be an Apollonian circle of A and A' ?

(e) In the DWEG model, if P and Q are D-points, explain how to construct the D-circle through Q with D-center P. Use your answers to earlier questions to make this brief but clear.

(f) Suppose a sphere is projected stereographically to the plane and that c is a circle on the sphere. If one spherical center X^ of c is projected to a point X and one point C^ on the circle is projected to a point C, explain how you could construct a circle in the plane that is the stereographic image of the circle c on the sphere. Use your answers to earlier questions to make this brief but clear.

Problem 3: Circles on the sphere.

A sphere s with center O intersects a plane p in a circle c. Suppose that the radius of s = 2 cm and the distance from O to p = 1 cm.

Answer these questions about c. Draw a sketch that illustrates your reasoning, show your work clearly and box your answers. The answers should provide an exact value (possibly using inverse trig functions). If it is possible to give an exact answer without inverse trig functions, that is even better. Decimal approximations may be useful as a check but are not required.

(a) What is the radius of c considered as a Euclidean circle in plane p?

(b) What is the spherical radius of c as a spherical circle on s (measured in degrees)?

Problem 4: Prove One of These Theorems

Prove one of these theorems. As always, you should be sure to state clearly at the beginning what you are proving. Also be very clear about stating definitions and any theorems you may be using in your proof.

(a) Tell what is the image of any line m when it is inverted in a circle c; then prove it.

(b) Given 3 circles, prove the radical axes of the 3 pairs of circles are concurrent (or parallel).

(c) Suppose the two circles c and d are orthogonal. If d is inverted in c, tell what is the image of d. Prove your assertion.

(d) On a sphere, prove that for any two great circles, there is exactly one great circle orthogonal to the two given circles.