Knowra Poncelet–Steiner theorem Poncelet–Steiner theorem The Poncelet–Steiner theorem states that every straightedge-and-compass construction can be performed using only an unmarked straightedge when a fixed circle and its center are given.
Polarization : In projective geometry, polarity is a correspondence between points and lines defined by a conic. The fixed circle supplies polar relationships that help construct intersections otherwise found with a compass.
Unmarked straightedge : A ruler-like instrument used to draw the line through two known points, without measuring distances. It is the only construction instrument permitted by the theorem.
Jean-Victor Poncelet : A French mathematician whose work helped establish projective geometry in the nineteenth century. The theorem bears Poncelet's name, though its standard proof is associated with Steiner.
Straightedge-and-compass construction : A classical construction method using an unmarked straightedge and a compass. The theorem reproduces its results only when the fixed circle and its center are supplied.
Harmonic range : A set of four collinear points is harmonic when its cross ratio equals minus one. Straightedge constructions of harmonic conjugates provide steps for recovering compass-equivalent constructions.
Euclidean geometry : The geometry of points, lines, angles, and distances in flat space. The theorem preserves the usual Euclidean construction goals despite restricting the instruments.
Jakob Steiner : A Swiss mathematician known for foundational work in geometry, including projective and synthetic geometry. Steiner gave the classical proof that the fixed circle and its center suffice.
Mohr–Mascheroni theorem : A theorem stating that compass-only constructions can reproduce every straightedge-and-compass construction. It reverses the instrument restriction, requiring a compass rather than a fixed circle and straightedge.
Complete quadrilateral : A figure formed by four lines, with their six pairwise intersection points. Its diagonal points enable straightedge-only constructions of harmonic conjugates.
Conic section : A curve obtained by intersecting a plane with a cone, including circles, ellipses, parabolas, and hyperbolas. The given circle is a special conic whose center supplies additional structure.
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