1. State the definition of an Apollonian circle of points A and B.
2. Circle the correct answer. If c is an Apollonian circle of points A and B, is B the inversion of A in c? Answer: (1) always (2) sometimes (3) never.
3. In the figure below, construct the Apollonian circle of A and B through P.

Given a Euclidean circle c and a Euclidean line m, let m' be the inversion of m in c. [You may include the point at infinity as a point of m if this simplifies your answers.]

1. If m passes through the center of c, what kind of object is m'?
2. If m does not pass through the center of c, what kind of object is m'?
3. Prove (b).

The figure below is a figure in the DWEG model, where the point O is the special point that is removed from the plane and is not a DWEG point. For your convenience the Euclidean circles x and y which are DWEG lines AB and AD have already been constructed (the Euclidean centers of x and y are X and Y).

The figure below shows the P-model inside circle c with two P-lines m and n (the ideal points EFGH are also in the figure).

1. Construct a P-line which is orthogonal (perpendicular) to both m and n.
2. If two P-lines p and q are critical parallels, is there a P-line orthogonal to both p and q? Tell whether this always, sometimes or never happens and give a clear explanation why.