Holditch's theorem
Encyclopedia
In plane geometry
, Holditch's theorem states that if a chord
of fixed length is allowed to rotate around a convex closed curve, then the locus
of a point on the chord a distance p from one end and a distance q from the other is a closed curve whose area is less than that of the original curve by
.
area is independent of the shape of the original curve, and that the area's form matches that for the area of an ellipse
. The theorem's author was a president of Caius College, Cambridge.
Plane geometry
In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...
, Holditch's theorem states that if a chord
Chord (geometry)
A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
of fixed length is allowed to rotate around a convex closed curve, then the locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
of a point on the chord a distance p from one end and a distance q from the other is a closed curve whose area is less than that of the original curve by
.Observations
The theorem is included as one of Clifford Pickover's 250 milestones. Some peculiarities of the theorem include that the area is independent of the shape of the original curve, and that the area's form matches that for the area of an ellipseEllipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
. The theorem's author was a president of Caius College, Cambridge.