Editing Poncelet–Steiner theorem (section)

==== Constructing a perpendicular line ====
[[File:3 - PerpendicularLine.png|thumb|right|250px|Construction of a perpendicular line.]]
This construction does require the use of the given circle and takes advantage of [[Thales's theorem]]. 

From a given line {{mvar|m}}, and a given point {{mvar|A}} in the plane, a [[perpendicular]] to the line is to be constructed through the point. Provided is the given circle {{math|''O(r)''}}.
# If the desired line from which a perpendicular is to be made, {{mvar|m}}, does not pass through the given circle - as is depicted - or it also passes through the given circle's center, then a new parallel line (in red) may be constructed arbitrarily such that it does pass through the given circle but not its center, and the perpendicular is to be made from this line instead.
# This red line which passes through the given circle but not its center, will intersect the given circle in two points, {{mvar|B}} and {{mvar|C}}.
# Draw a line {{math|{{overline|''BO''}}}} (in orange), through the circle center.
#* This line intersects the given circle at point {{mvar|D}}.
#* Angle {{math|∠''BOD''}} is 180°.
# Draw a line {{math|{{overline|''DC''}}}} (in light green).
#* This line is perpendicular to the red (and therefore the black) lines, {{math|{{overline|''BC''}}}} and {{mvar|''m''}}.
#* By Thales's Theorem, angle {{math|∠''BCD''}} is 90°.
# Construct a parallel of line {{math|{{overline|''DC''}}}} through point {{mvar|''A''}} using previous constructions.
#* A perpendicular of the original black line, {{mvar|''m''}}, now exists in the plane, line {{math|{{overline|''DC''}}}}.
#* A parallel of any line may be constructed through any point in the plane.

If the line from which a perpendicular is to be made does pass through the circle center, an alternative approach would be to construct the tangent lines to the circle at the lines points of intersection, using Steiner constructions. This is not demonstrated in this article.

Another option in the event the line passes through the circle's center would be to construct a parallel to it through the circle at an arbitrary point.  An [[isosceles trapezoid]] (or potentially an isosceles triangle) is formed by the intersection points to the circle of both lines.  The two non-parallel sides of which may be extended to an intersection point between them, and a line drawn from there through the circle's center.  This line is perpendicular, and the diameter is bisected by the center.

By an alternative construction not demonstrated in this article, a perpendicular of any line may be constructed without a circle, provided there exists in the plane any [[square]].