You will have Sketchpad scripts on the server to construct perpendicular bisectors in the P model. (You did this for yourself last Friday, so you may already have such a tool.)

Draw 3 points A, B, C in the P-model. Construct the perpendicular bisectors of the sides of triangle ABC (in the P-model!). Drag A, B, C around and observe whether the perpendicular lines are concurrent. More specifically,

1. If two of the perpendicular bisectors intersect at a point Q in the P-model, then are all 3 lines concurrent?
2. If two of the perpendicular bisectors are limiting parallel, i.e., they intersect at a point Q at infinity in the P-model, then are all 3 lines limiting parallel at this point?
3. If each pair of the perpendicular bisectors is ultraparallel, do the 3 lines lie in a hypebolic pencil? Check this by constructing a P-line orthogonal to two of the lines and see whether this is orthogonal to the other two lines. One way to check this is to check some features: (1) are the Euclidean centers of the support circles of the 3 perpendicular bisectors collinear? (2) Is the radical axis of each pair the same Euclidean line? (3) Check how that the 3 radical axes of each perpendicular line with the "horizon" circle in the P-model are all concurrent.
4. Construct the Euclidean circle through A, B, C. In each case of (a), (b), (c) above, when is this circle a P-circle, a P-horocycle, or other?
5. Does this conclusion seem to be valid? For any triangle ABC in the P-model, the 3 perpendicular bisectors of the sides always belong to a pencil.

Study the internal and external bisectors of a P-triangle ABC. In Euclidean geometry the bisectors concur 3 at a time at 4 points. In the P-model, do these 4 points seem to be replaced by 4 pencils, based on your experiments?

You will be given a script constructing the antipodal point P’ of a point P. To use the script you have to have a reference circle in the plane representing the image of the equator. (This will be explained in class.)

All these are tasks in spherical geometry which should be done on the image of the sphere on the plane obtained by stereographic projection.

Task 1. Given two points A and B construct the great circle through A and B.

Task 2. Given a point A and a great circle c, construct the great circle through A that is orthogonal to c.

Task 3. Given a great circle c, construct the points C1 and C2 that are the poles of c.

Task 4. Construct a 90-90-90 triangle which does not have the north or south pole as a vertex.

Task 5. Given two points A and B, construct the perpendicular bisector of AB.