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We also discuss cab-equidistant paths from point A to point B, as the concept of equal cab-distance will be important when we investigate what the circle and the perpendicular bisector in taxicab geometry look like.
As the answers are quite unexpected, the students proceed cautiously: it is very surprising for students to find out that the locus of points equidistant from a center looks like a square and that the perpendicular bisector is not a straight line.
Construct the perpendicular bisector of AF and mark the point M where it cuts AF.
Draw a line through A parallel to DF to cut the perpendicular bisector plane at P.
Because MP is the perpendicular bisector of AF, the radius of the sphere FP is equal to the length of the cylinder AP.
One might begin with simple results, such as folding a perpendicular bisector enables the construction of an isosceles triangle.