弦 定义
A chord of a circle is a straight line segment on the interior of a circle whose endpoints both lie on that circle. The infinite line extension of a chord is a secant line, or just secant. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word chord is derived from the Latin term chorda meaning bowstring. The term is also used in graph theory, where a cycle chord of a graph cycle C is an edge not in C whose endpoints lie in C.
界
圆的和弦的性质如下:
当且仅当它们的长度相等时，和弦到中心的距离是等距的。
相等的弦从圆心被相等的角度对着。
通过圆心的弦称为直径，是最长的弦。
如果弦 AB 和 CD 的延长线（割线）相交于点 P，则它们的长度满足 AP·PB = CP·PD（点的幂定理）。
椭圆
椭圆的一组平行弦的中点是共线的。
三角学
和弦在三角学的早期发展中被广泛使用。由 Hipparchus 编制的第一个已知三角表将每 7.5degrees 的弦函数值制成表格。公元二世纪，亚历山大的托勒密在他的天文学著作中编制了一个更广泛的和弦表，以半度为增量给出了从 1/2 度到 180 度的角度的弦值。圆的直径为 120，弦长精确到整数部分后的两位以 60 为基数的数字。
The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length: crd θ = √ (1 – cos θ)2 + sin2 θ = √ 2 – 2cos θ = 2 sin(θ⁄2).
最后一步使用半角公式。就像现代三角学建立在正弦函数上一样，古代三角学建立在和弦函数上。据说喜帕恰斯写了一部十二卷的和弦作品，现在都丢失了，所以大概对它们有很多了解。在下表中（其中 c 是弦长，D 是圆的直径），弦函数可以满足许多类似于现代著名的恒等式：
姓名 |
基于正弦 |
基于和弦 |
毕达哥拉斯 |
sin2 θ + cos2 θ = 1 |
crd2 θ + crd2 (π - θ) = 4 |
半角 |
sin θ⁄2 = ± √ 1 - cos θ⁄2 |
crd θ⁄2 = ± √ 2 - crd(π - θ) |
Apothem (一) |
c = 2√ r2 - a2 |
c = √ D2 - 4a2 |
角度 (θ) |
c = 2r sin(θ⁄2) |
c = D⁄2crd θ |
相关定义
来源
“Chord.” From Wolfram MathWorld, mathworld.wolfram.com/Chord.html.
“Chord (Geometry).” Wikipedia, Wikimedia Foundation, 7 May 2020, en.wikipedia.org/wiki/Chord_(geometry).