从垂直到垂直
百度几何吧转载https://tieba.baidu.com/p/7451652229 本帖最后由 dlsh 于 2021-12-17 19:20 编辑
贴吧网友qzc的解答：
只要证明这个结论：
S_1S_2S_3是S关于△ABC的塞瓦三角形,BU,CV分别平行S_1S_2,S_1S_3交对边于U,V.UV交BC于T,则AT平行S的三线性极线.
证明：设BU∩CV=W，G为△ABC的重心，作过A,B,C,G,S的二次曲线Γ，设A,B,C关于Γ的极点分别为A',B',C'，则A'在S_2S_3上等.下面证明W∈B'C'.
让S在Γ上运动,则BU∩B'C'→C'S_2→BS→S→CV∩B'C'是射影对应,则只要考虑三种情况使它们相等.
当S=G时显然成立.
当S=C时,BU∥CC',故BU=BB',CV∥CB',故CU=CB',此时结论成立
当S=B时同上可知结论成立
设B'C'∩BC=T*,则由W∈B'C'知=-1,故A,A',T共线,故AT和S的三线性极线平行.
看不懂，图片链接不出来，只好上传。 如果先构造P，再作R点，利用Ceva定理，出现高次方程，先用Ceva定理，求出三边比例 关系，出现Ceva点不同,十分费解。
(*Clear["Global`*"]*)
\!\(\*OverscriptBox["a", "_"]\) = 1/a;
\!\(\*OverscriptBox["b", "_"]\) = 1/b;
\!\(\*OverscriptBox["c", "_"]\) = 1/c;(*
\!\(\*OverscriptBox["p", "_"]\)=-((-a+2 b-c-p)/(a c ))*);
k := (a - b)/(
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\));
\!\(\*OverscriptBox["k", "_"]\) := 1/k;(*复斜率定义*)
\!\(\*OverscriptBox["Jd", "_"]\) := -((a1 - k1
\!\(\*OverscriptBox["a1", "_"]\) - (a2 - k2
\!\(\*OverscriptBox["a2", "_"]\)))/(
k1 - k2));(*复斜率等于k1,过点A1与复斜率等于k2,过点A2的直线交点*)
Jd := -((k2 (a1 - k1
\!\(\*OverscriptBox["a1", "_"]\)) - k1 (a2 - k2
\!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
Chuizu := (
\!\(\*OverscriptBox["a", "_"]\) b - a
\!\(\*OverscriptBox["b", "_"]\) + p (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) +
\!\(\*OverscriptBox["p", "_"]\) (a - b))/(2 (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)));(*=(1/2)[p+(
\!\(\*OverscriptBox["a", "_"]\)b-a
\!\(\*OverscriptBox["b", "_"]\)+
\!\(\*OverscriptBox["p", "_"]\)(a-b))/(
\!\(\*OverscriptBox["a", "_"]\)-
\!\(\*OverscriptBox["b", "_"]\))]P到直线AB的垂足*)
\!\(\*OverscriptBox["Chuizu", "_"]\) := (a
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["a", "_"]\) b +
\!\(\*OverscriptBox["p", "_"]\) (a - b) + p (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)))/(2 (a - b));
FourPoint := ((
\!\(\*OverscriptBox["c", "_"]\) d - c
\!\(\*OverscriptBox["d", "_"]\)) (a - b) - (
\!\(\*OverscriptBox["a", "_"]\) b - a
\!\(\*OverscriptBox["b", "_"]\)) (c - d))/((a - b) (
\!\(\*OverscriptBox["c", "_"]\) -
\!\(\*OverscriptBox["d", "_"]\)) - (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) (c - d));(*过两点A和B、C和D的交点*)
\!\(\*OverscriptBox["FourPoint", "_"]\) := -(((c
\!\(\*OverscriptBox["d", "_"]\) -
\!\(\*OverscriptBox["c", "_"]\) d) (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) - ( a
\!\(\*OverscriptBox["b", "_"]\) -
\!\(\*OverscriptBox["a", "_"]\) b) (
\!\(\*OverscriptBox["c", "_"]\) -
\!\(\*OverscriptBox["d", "_"]\)))/((a - b) (
\!\(\*OverscriptBox["c", "_"]\) -
\!\(\*OverscriptBox["d", "_"]\)) - (
\!\(\*OverscriptBox["a", "_"]\) -
\!\(\*OverscriptBox["b", "_"]\)) (c - d)));
f = Chuizu;
\!\(\*OverscriptBox["f", "_"]\) =
\!\(\*OverscriptBox["Chuizu", "_"]\); e = Chuizu;
\!\(\*OverscriptBox["e", "_"]\) =
\!\(\*OverscriptBox["Chuizu", "_"]\); d = Chuizu;
\!\(\*OverscriptBox["d", "_"]\) =
\!\(\*OverscriptBox["Chuizu", "_"]\);
r1 = FourPoint;
\!\(\*OverscriptBox["r1", "_"]\) =
\!\(\*OverscriptBox["FourPoint", "_"]\); r =
FourPoint;
\!\(\*OverscriptBox["r", "_"]\) =
\!\(\*OverscriptBox["FourPoint", "_"]\);
u = Jd[-k, b, -a c, a];
\!\(\*OverscriptBox["u", "_"]\) =
\!\(\*OverscriptBox["Jd", "_"]\)[-k, b, -a c, a]; v =
Jd[-k, c, -a b, a];
\!\(\*OverscriptBox["v", "_"]\) =
\!\(\*OverscriptBox["Jd", "_"]\)[-k, c, -a b, a];
t = FourPoint;
\!\(\*OverscriptBox["t", "_"]\) =
\!\(\*OverscriptBox["FourPoint", "_"]\);
Simplify[{r, r1, , r1 - r}](*验证Ceva点*)
Factor[{1, r, r1, , r1 - r}]
Simplify[{2, u,
\!\(\*OverscriptBox["u", "_"]\), , v, v, , t, t, , k, k, ,
k - k}](*k,k+k*)
Factor[{2, u,
\!\(\*OverscriptBox["u", "_"]\), , v, v, , t, t, , k,
k}](*k,k+k*)
表达式太长，就不上传了，感觉不对。
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