论坛 › 数学欣赏 › 史勇的一个经典几何案例

dlsh 发表于 2022-1-25 21:45:46

史勇的一个经典几何案例

乘积与线段平方的差
https://imgsa.baidu.com/forum/w%3D580/sign=61bf0561dfc8a786be2a4a065708c9c7/f0974ced2e738bd4f7bc17aaaf8b87d6267ff9ab.jpg
http://imgsrc.baidu.com/forum/pic/item/bbaca6ec08fa513d4237c945336d55fbb3fbd9ab.jpg

markfang2050 发表于 2022-1-27 14:17:09

凡是以自己名字标榜的所谓定理发现，都是“妄人”。整天搞这些初等数学题，真想不通不是在浪费生命？真有能力，真知灼见，还是发表论文或搞一些应用项目也好。对国家社会个人也是好事。

TSC999 发表于 2022-1-28 22:34:35

史勇不知是何方大侠？ 他有一套奇特的对合理论，对于平面几何图形的构图方法有独到研究。他在机器证明中好像很少用到共轭复数。

彭翕成 发表于 2022-1-31 10:26:42

很难想象这里的E，F，坐标能很简单的求出来

dlsh 发表于 2022-2-1 20:24:35

Clear["Global`*"]

\!\(\*OverscriptBox["o", "_"]\) = o = 0;
\!\(\*OverscriptBox["a", "_"]\) = 1/a;
\!\(\*OverscriptBox["b", "_"]\) = 1/b; b = -I/v; c = -
\!\(\*OverscriptBox["b", "_"]\);
\!\(\*OverscriptBox["c", "_"]\) = 1/c;(*圆心在原点，假设e^(i\)=v*)
m = (b + c)/2;
\!\(\*OverscriptBox["m", "_"]\) = (
\!\(\*OverscriptBox["b", "_"]\) +
\!\(\*OverscriptBox["c", "_"]\))/2; h = a + b + c;
\!\(\*OverscriptBox["h", "_"]\) =
\!\(\*OverscriptBox["a", "_"]\) +
\!\(\*OverscriptBox["b", "_"]\) +
\!\(\*OverscriptBox["c", "_"]\);(*外心在原点成立*)
k := (a - b)/(
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\));
\!\(\*OverscriptBox["k", "_"]\) := 1/k;(*复斜率定义*)

e = a ((a - c) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) - (b - c) (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) )/(b - a) + b; f = a ((a - b) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) - (b - c) (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)))/(a - c) + c;(*把
\!\(\*OverscriptBox["BA", "\"]\)和
\!\(\*OverscriptBox["BE", "\"]\)旋转到
\!\(\*OverscriptBox["AC", "\"]\)相同的方向，再利用复斜率，根据线段相等条件求得*)

\!\(\*OverscriptBox["e", "_"]\) =
\!\(\*OverscriptBox["a", "_"]\) ((a - c) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) - (b - c) (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)))/(
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["a", "_"]\)) +
\!\(\*OverscriptBox["b", "_"]\);
\!\(\*OverscriptBox["f", "_"]\) =
\!\(\*OverscriptBox["a", "_"]\) ((a - b) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) - (b - c) (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)))/(
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) +
\!\(\*OverscriptBox["c", "_"]\);
n = (e + f)/2;
\!\(\*OverscriptBox["n", "_"]\) = (
\!\(\*OverscriptBox["e", "_"]\) +
\!\(\*OverscriptBox["f", "_"]\))/2;
Simplify[{b,
\!\(\*OverscriptBox["b", "_"]\), c,
\!\(\*OverscriptBox["c", "_"]\), , e,
\!\(\*OverscriptBox["e", "_"]\), f,
\!\(\*OverscriptBox["f", "_"]\)}]
Simplify[{h, m, n, k, k, , (h - o)/(m - n), (
\!\(\*OverscriptBox["h", "_"]\) -
\!\(\*OverscriptBox["o", "_"]\))/(
\!\(\*OverscriptBox["m", "_"]\) -
\!\(\*OverscriptBox["n", "_"]\))}]



\!\(\*OverscriptBox["b", "_"]\) = b = 0;
\!\(\*OverscriptBox["c", "_"]\) = c = 1; a = 1/(1 - \ v);
\!\(\*OverscriptBox["a", "_"]\) = v/(v - \);(*假设
\!\(\*OverscriptBox["AC", "\"]\)/
\!\(\*OverscriptBox["AB", "\"]\)=\v*)
m = (b + c)/2;
\!\(\*OverscriptBox["m", "_"]\) = (
\!\(\*OverscriptBox["b", "_"]\) +
\!\(\*OverscriptBox["c", "_"]\))/2; n = (e + f)/2;
\!\(\*OverscriptBox["n", "_"]\) = (
\!\(\*OverscriptBox["e", "_"]\) +
\!\(\*OverscriptBox["f", "_"]\))/2;
k := (a - b)/(
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\));
\!\(\*OverscriptBox["k", "_"]\) := 1/k;(*复斜率定义*)
kAB = k; kAC = k;
Chuixin := (
\!\(\*OverscriptBox["a", "_"]\) (b - c) (b + c - a) +
\!\(\*OverscriptBox["b", "_"]\) (c - a) (c + a - b) +
\!\(\*OverscriptBox["c", "_"]\) (a - b) (a + b - c) )/((b - c)
\!\(\*OverscriptBox["a", "_"]\) + (-a + c)
\!\(\*OverscriptBox["b", "_"]\) + (a - b)
\!\(\*OverscriptBox["c", "_"]\));(*垂心公式*)

\!\(\*OverscriptBox["Chuixin", "_"]\) := -((a (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) (
\!\(\*OverscriptBox["b", "_"]\) +
\!\(\*OverscriptBox["c", "_"]\) -
\!\(\*OverscriptBox["a", "_"]\)) + b (
\!\(\*OverscriptBox["c", "_"]\) -
\!\(\*OverscriptBox["a", "_"]\)) (
\!\(\*OverscriptBox["c", "_"]\) +
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) + c (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) (
\!\(\*OverscriptBox["a", "_"]\) +
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) )/((b - c)
\!\(\*OverscriptBox["a", "_"]\) + (-a + c)
\!\(\*OverscriptBox["b", "_"]\) + (a - b)
\!\(\*OverscriptBox["c", "_"]\)));
Waixin := (a
\!\(\*OverscriptBox["a", "_"]\) (b - c) + b
\!\(\*OverscriptBox["b", "_"]\) (c - a) + c
\!\(\*OverscriptBox["c", "_"]\) (a - b) )/(
\!\(\*OverscriptBox["a", "_"]\) (b - c) +
\!\(\*OverscriptBox["b", "_"]\) (c - a) +
\!\(\*OverscriptBox["c", "_"]\) (a - b));
\!\(\*OverscriptBox["Waixin", "_"]\) := -((a
\!\(\*OverscriptBox["a", "_"]\) (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) + b
\!\(\*OverscriptBox["b", "_"]\) (
\!\(\*OverscriptBox["c", "_"]\) -
\!\(\*OverscriptBox["a", "_"]\)) + c
\!\(\*OverscriptBox["c", "_"]\) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) )/(
\!\(\*OverscriptBox["a", "_"]\) (b - c) +
\!\(\*OverscriptBox["b", "_"]\) (c - a) +
\!\(\*OverscriptBox["c", "_"]\) (a - b)));(*外心公式*)

h = Chuixin;
\!\(\*OverscriptBox["h", "_"]\) =
\!\(\*OverscriptBox["Chuixin", "_"]\); o = Waixin;
\!\(\*OverscriptBox["o", "_"]\) =
\!\(\*OverscriptBox["Waixin", "_"]\);

e = kAC ((a - c) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) - (b - c) (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) )/((b - a) I v) + b; f =
v kAB ((a - b) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) - (b - c) (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)))/(I (a - c)) + c;(*把
\!\(\*OverscriptBox["BA", "\"]\)和
\!\(\*OverscriptBox["BE", "\"]\)旋转到
\!\(\*OverscriptBox["AC", "\"]\)相同的方向，再利用复斜率，根据线段相等条件求得*)

\!\(\*OverscriptBox["e", "_"]\) = v ((a - c) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) - (b - c) (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)))/(-I (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["a", "_"]\)) kAC) +
\!\(\*OverscriptBox["b", "_"]\);
\!\(\*OverscriptBox["f", "_"]\) = ((a - b) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) - (b - c) (
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)))/(-I (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\)) v kAB) +
\!\(\*OverscriptBox["c", "_"]\);

Simplify[{e,
\!\(\*OverscriptBox["e", "_"]\), f,
\!\(\*OverscriptBox["f", "_"]\)}]
Simplify[{h,
\!\(\*OverscriptBox["h", "_"]\), o,
\!\(\*OverscriptBox["o", "_"]\), , m,
\!\(\*OverscriptBox["m", "_"]\), n,
\!\(\*OverscriptBox["n", "_"]\), , k, k}]
Simplify[{(h - o)/(n - m)}]

dlsh 发表于 2022-2-2 22:34:07

\(设B在原点，c=1,e^{i\alpha}=u{,}e^{i\beta}=v{,}\)，染色部分的几何意义如图中的数据，这种构图比前面两种对称

TSC999 发表于 2022-2-3 21:24:28

TSC999 发表于 2022-2-3 21:33:10

现在的问题是，史勇关于 E、F 点的复坐标公式是怎样算出来的？ 是根据 “复对合理论”算的？“复对合” 是个啥东东？ 网上没见到史勇先生出版过这个理论的书。
不知哪位大师也能算出E、F 点的复坐标？ 搬个板凳坐在这里等。

彭翕成 发表于 2022-2-4 23:07:29

TSC999 发表于 2022-2-3 21:24


根据几何意义可知，BC弧中点就是复数bc。再根据等腰三角形三线合一，可知c方-b方与bc垂直。群里几个人已经建群，如果你愿意加入，可加我微信13720152511，方便交流一些

dlsh 发表于 2022-2-5 22:12:38

把BC最小和线段乘积等于平方差改一下，类似的线段可能有三条

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