Given a circle c and a line m, prove that the inversion of m in c is either a circle or a line. (The point at infinity is considered to be included in all lines).

1. A triangle ABC in the P-model has 3 perpendicular bisectors of the sides. State the possible relationships among these 3 lines and draw a sketch illustrating each of the possibilities.

Is this possible? If so, sketch an example.

2. Is there a set of 3 points in the P-model which are not P-collinear and also not contained in a P-circle. Explain. Make a sketch.

1. In the figure is a line AB and a point P on this line. Construct with straightedge and compass a point Q so that PQ divides AB harmonically.
2. On the number line, let the points A, B, C, D be 0, 100, 99, 101. What is the cross ratio R(A, B, C, D)? Use cross ratio to tell whether A, B, C, D is harmonic.

Write an explanation that tells what inversion is and the relationship between inversion and orthogonal circles. Be clear about key concepts and reasons. Imagine that your audience is a student who has just taken Math 444 and thus knows the basic theorems of plane geometry but has never heard of inversion.

In the figure are two points A and B, a segment CD of length d and a line m. The set of P such that |AP| + |BP| = d is an ellipse. Construct the point P of the ellipse which is on line m such that m is the tangent line to the ellipse at P.

State the Pascal theorem about conics and explain briefly how this is used to construct a conic through 5 points..

Draw a figure with two triangles A1 B1 C1 and A2 B2 C2 that illustrates the Desargues Theorem when the intersection of A1 A2 and B1 B2 is a point at infinity (i.e., the two coplanar triangles are perspective from a point at infinity).

Consider a figure made up of 3 lines OA1, OA2 OA3 and 3 circles through O with diameters OA1, OA2 OA3. Invert in circle with center O. What does image look like? Sketch the image. Indicate important relationships.