Editing Poncelet–Steiner theorem (section)

==== Rotating a line segment ====
Line segment rotation is a useful construction but may rarely be used in practice. It is necessary to demonstrate in order to fully generalize the theorem.

To define a circle only the center and one point - any point - on the circumference is required.  In principle a new point {{mvar|''B' ''}} is constructed such that circle {{mvar|''A(B)''}} is equal to circle {{mvar|''A(B')''}}, though the point {{mvar|''B''}} is not equal to point {{mvar|''B' ''}}. In essence, segment {{math|{{overline|''AB''}}}} is rotated about the axis point {{math|''A''}}, to {{math|{{overline|''AB' ''}}}}, for a different set of defining points for the same circle. 

[[File:Segment rotation.png|400px|thumb|Steiner construction of the rotation of a line segment]]

One way of going about this which satisfies most conditions is as follows:
# Draw the line segment {{math|{{overline|''AB''}}}} (in black).
# Construct a parallel (in red) of line {{math|{{overline|''AB''}}}} through the center, point {{mvar|''O''}}, of the given circle.
#* The parallel intersects the given circle at some point {{mvar|''b''}}.
# Choose a point {{math|''b' ''}} arbitrarily on the given circle not colinear with line {{math|{{overline|''Ob''}}}}.
# Draw a line {{math|{{overline|''Ob' ''}}}} (in orange).
# Construct a parallel line to {{math|{{overline|''Ob' ''}}}} through point {{math|''A''}} (in magenta).
# Draw a line {{math|{{overline|''bb' ''}}}} (in light green), connecting the points on the circles circumference.
# Construct a parallel of line {{math|{{overline|''bb' ''}}}} through point {{math|''B''}} (in blue).
# Intersect the blue and pink parallels, from points {{math|''B''}} and {{math|''A''}}, respectively.
#* This is point {{math|''B' ''}}.
#* Point {{mvar|''B' ''}} is the desired point, rotating the line segment and defining the same circle centered at {{math|''A''}}.

This construction will fail if the desired rotation is [[diametrically opposite]] the circle (i.e. a half-circle rotation). One solution to this scenario is to employ two separate rotation constructions, neither one a half-circle rotation from the previous, one acting as an intermediary step. Choose any rotation angle arbitrarily, complete the rotation, then choose the [[supplementary angle]] and perform the rotation a second time.

There does exist a second, alternative rotation construction solution, based on projections and perspective points. Though it avoids the aforementioned half-circle rotation complication, it does have its own complications, which are similarly resolved with intermediary rotation constructions. The construction is no more versatile. It is not demonstrated in this article.