Problems 5.1 – 5.4 are stated in Lab 5.
For this assignment you can assume that for two points A and B in 3-space, the set of points equidistant from A and B is a plane M orthogonal to line AB through the midpoint of AB, called the perpendicular bisecting plane. Also, reflection of a point A in a plane Q is defined to be the point A' such that Q is the p perpendicular bisecting plane of AA'.
These questions are all about geometry on a sphere with center O and radius R; all distances, triangles, etc. are spherical unless specified to the contrary. In the proofs, try to do your best job of clear thinking, legible writing, and good illustrations.
5.5 Spherical perpendicular bisector
(a) Given two points A and B on the sphere, prove that the set of points equidistant from A and B is a great circle m.
(b) If c is a great circle through A and B, at what points does the great circle m above intersect c? What does this say about the concept of a midpoint of a segment on the sphere?
(c) If c is a great circle through A and B, prove that c is orthogonal to the great circle m above.
Definition: The (spherical) perpendicular bisector of AB is the great circle m that is the set of points equidistant from A and B.
5.6 Concurrence
(a) Prove that the perpendicular bisectors of the sides of a spherical triangle ABC are concurrent at a point P. (Is P the only point?)
(b) Prove that for any spherical triangle, there is a circle d that passes through A, B, and C.
5.7 Line Reflection
(a) If d is a great circle and A is a point on the sphere, explain why the reflection of A in the plane containing d is a point A' that is also on the sphere.
(b) If is true that a great circle e through AA' is orthogonal to d? Why?
Suppose the sphere is in (x,y,z) space with center (0,0,0), and p1, p2, p3 are the great circles that are intersections of the sphere with the coordinate planes, x = 0, y = 0, z = 0.
(c) Describe clearly each of the transformations that are the composition of reflection in two of these great circles.
(d) Describe clearly each of the transformations that are the composition of reflection in three of these great circles