Consider a segment BC and a reflection R across a line l on a cylinder. Assume that the reflection exchanges B and C, that is, R(B) = C and R(C) = B.

(a) If l is a vertical line, does l necessarily intersect BC? How about if l is a (horizontal) great circle? Give a proof or a counterexample in each case.

(b) If l is a vertical line, does R necessarily take BC to itself? How about if l is a great circle? Give a proof or a counterexample in each case.

(c) If l is a vertical line and l does intersect BC at a point D, is l necessarily the perpendicular bisector of BC? How about if l is a great circle? Give a proof or a counterexample in each case.

(d) Now consider a triangle ABC so that AB AC and l is the angle bisector of angle BAC. If l is a vertical line, is the Isosceles Triangle Theorem true for triangle ABC? The Isosceles Bisector Theorem? How about if l is a great circle? Give an explanation or a counter example in each case using your answers above.