Properties of Chords in Circles
– Chords are equidistant from the center if their lengths are equal.
– Equal chords are subtended by equal angles from the center of the circle.
– A chord passing through the center of a circle is called a diameter, which is the longest chord.
– If the line extensions of two chords intersect at a point, their lengths satisfy a specific equation (power of a point theorem).
– The midpoints of a set of parallel chords of a conic are collinear (midpoint theorem for conics).
Chords in Trigonometry
– Chords were extensively used in the early development of trigonometry.
– Hipparchus compiled the first known trigonometric table, which tabulated the value of the chord function for specific angles.
– Ptolemy compiled a more extensive table of chords, accurate to two sexagesimal digits, in his book on astronomy.
– The chord function is geometrically defined as the length of the chord between two points on a unit circle separated by a central angle.
– The chord function can be related to the modern sine function using the Pythagorean theorem.
Types of Chords and Functions
– Circular segment is the part of the sector that remains after removing the triangle formed by the center and the two endpoints of the circular arc.
– Scale of chords is a table of chord lengths for specific angles.
– Exsecant and excosecant are trigonometric functions related to chords.
– Versine and haversine are functions related to chords, with specific formulas.
– Zindler curve is a closed and simple curve where all chords dividing the arc length into halves have the same length.
Notable Concepts and Paradoxes
– Holditch’s theorem relates to a chord rotating in a convex closed curve.
– Circle graph is a graphical representation using chords.
– Bertrand paradox is a probability paradox related to the average length of chords.
– Chakerian’s book ‘A Distorted View of Geometry’ discusses various geometric concepts, including chords.
– Maor’s book ‘Trigonometric Delights’ explores the history and applications of trigonometry, including chords.
Additional Resources
– ‘On the Shoulders of Giants: The Great Works of Physics and Astronomy’ edited by S.W. Hawking is a recommended book that may cover the topic of chords.
– Jiří Stávek’s article ‘On the hidden beauty of trigonometric functions’ discusses the aesthetic aspects of trigonometric functions, including chords.
– Wikimedia Commons has media related to chords in geometry.
– ‘On Sizes and Distances’ is a problem related to chords in Aristarchus’ work.
– The Canadian Center of Science and Education may have additional resources on the topic of chords. Source: https://en.wikipedia.org/wiki/Chord_(geometry)
A chord (from the Latin chorda, meaning "bowstring") of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a secant line. The perpendicular line passing through the chord's midpoint is called sagitta (Latin for "arrow").
More generally, a chord is a line segment joining two points on any curve, for instance, on an ellipse. A chord that passes through a circle's center point is the circle's diameter.