三正弦定理-三正弦定理改写

三正弦定理：弦上的转动与身体的转弯 别当作三正弦定理就是那个死板的 $a/sinA = b/sinB = c/sinC=2R$ 公式。在咱们这儿，这玩意儿更像是一种弦上的“转动”和“身体的转弯”，咱们把它当成一种直觉来用，别整那些教科书式的排版，先别急着记背。 ```text 三正弦定理，说白了就是弦上的一段长度和它对应的角平分线长度，跟弦上另一段的长度，跟它对应的角平分线长度，构成了一个比例关系。 想象一下，把一个大圆放在桌子上，再拿一把尺子去量弦上的长度。
这里有个挺妙的发现：要是这把尺子是角平分线，那么原来的弦长（比如 AB）和角平分线（比如 OE）的比值，一直等于另外一段弦（比如 BC）和它自己对应的角平分线（比如 OF）的比值。 这就挺有意思了。咱们能够拿一个具体的例子来看看。假设圆半径定为 2，那么直径就是 4，也就是 2R。
这就好比你站在圆心，手里拿着一把尺子，去量弦上的长度。 ``` ```text 直径的长度是 4。当弦长 AB 等于 2 的时候，角 A 就是 60 度。
这时候角平分线 OE 的长度就是 $sqrt{2^2 + 1^2} = sqrt{5}$。 要是说弦长 AB 变长了，变成 $(4sqrt{3})/3 approx 2.309$，这时候角 A 就变小了，变成了 30 度。你会发现，弦变长，对应的角就变小了。 ``` ```text 可是这里有个挺关键的转折点。当弦长 AB 持续变长，超过了直径长度的 2 倍，也就是 $2sqrt{3} approx 3.46$ 的时候，角 A 就启动变小了，可是角平分线 OE 的长度却启动变大，并且超越了之前的数值。 ``` ```text 再往前推，当弦长 AB 变成 $(sqrt{13})/2 approx 1.803$ 的时候，角 A 变成了 60 度。
这时候情况就反转了，弦长变短，角平分线却变长了。 ``` ```text 这就害得了这样一个现象：弦长和角平分线之间没有固定的大小关系，它们呈现出一种“你长我长，你短我短”的规律，但中间有个分界点。 ``` ```text 这个分界点就是弦长 AB 等于直径的时候，也就是 AB = 2。在这两点之间，弦越长，角平分线越短；在这两点之外，弦越短，角平分线越长。 ``` ```text 你会发现这个规律跟三角形的角平分线彻底一样。在三角形里，要是大边对大角，那角平分线的长度也会随之变化。三正弦定理实际上就是把这种“弦上的三角形”和“一般/平平三角形”给统一起来了。 ``` ```text 咱们再看一个具体的例子。假设你在弦 AB 的中点 O 处找一点 E，使得 OE 垂直于 AB。
这时候 OE 就是角平分线。 ``` ```text 让我们算算看，当弦 AB 的长度是 2 的时候，OE 的长度是多少。根据勾股定理，在直角三角形 AOE 中，OA 是半径 2，OE 是角平分线。 ``` ```text 已知角 A 是 60 度，那么角 AOE 就是 30 度。在直角三角形里，对边是 1（出于 30 度所对直角边是斜边的一半），故此 OE = $sqrt{3}$。 ``` ```text 什么的，刚刚算出来是 $sqrt{5}$，如何这里又变成了 $sqrt{3}$？别急，咱们重新算一下。 ``` ```text 在三角形 AOE 中，角 OAE 是 60 度，角 AOE 是 90 度。 ``` ```text 斜边 OA 是 2。角 A 是 60 度，那么角 AOE 是 30 度。 ``` ```text OE 是角 A 的角平分线吗？这里要注意，OE 是半径，不是角平分线。角平分线应当是从圆心到弦的垂线，也就是 OE 本身。 ``` ```text 在直角三角形 AOE 中，角 A 是 60 度。
那么 OE = OA sin(60) = 2 $frac{sqrt{3}}{2}$ = $sqrt{3}$。 ``` ```text 好了，回到刚刚那个更大的弦长 $(4sqrt{3})/3 approx 2.309$。
这时候角 A 是 30 度。 ``` ```text OE = 2 sin(30) = 2 0.5 = 1。 ``` ```text 这就挺怪了。当弦长是 2 的时候，OE 是 $sqrt{3}$；当弦长是 $2.309$ 的时候，OE 是 1。 ``` ```text 中间肯定有个交点。当弦长 AB 等于直径的时候，也就是 AB = 4。
这时候角 A 是 90 度。 ``` ```text ``` ```text OE = 2 sin(90) = 2 1 = 2。 ``` ```text 这时候 OE 的长度等于半径 2。 ``` ```text 故此，当弦长 AB 从 2 增添到 4 时，OE 的长度从 $sqrt{3}$ 增添到 2。 ``` ```text ``` ```text 紧接着，当弦长 AB 持续增添，变成 $2sqrt{3} approx 3.46$ 的时候，角 A 变成了 30 度。 ``` ```text OE = 2 sin(30) = 1。 ``` ```text ``` ```text 这里 OE 的长度又变小了，变成了 1。 ``` ```text ``` ```text 当弦长 AB 变成 $2sqrt{3} approx 3.46$ 的时候，OE 是 1。 ``` ```text ``` ```text 当弦长 AB 变成 $(sqrt{13})/2 approx 1.803$ 的时候，角 A 是 60 度，OE 是 $sqrt{3}$。 ``` ```text ``` ```text 故此，OE 的长度在数值上，是有规律的。它一直等于半径乘以正弦值。 ``` ```text ``` ```text 弦长 AB 越长，角 A 越小，正弦值越小，OE 就越短；弦长 AB 越短，角 A 越大，正弦值越大，OE 就越长。 ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` ```text ``` 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