蝴 蝶 定 理 复 数 法 证 明

M 是 AB 弦的中点，求证 PM=QM

Clear["Global`*"];(*令 ABCDEF 都在单位圆上且 AB 平行于实轴，设DE的复斜率为u，CF的复斜率为v*)
(OverBar[a])=1/a; (OverBar[b])=1/b;
Jd[a_,b_,c_,d_]:=((a b(c+ d)-c d(a+b))/(a b-c d));(OverBar[Jd])[a_,b_,c_,d_]:=((a+b-c-d)/(a b-c d)); (*单位圆上AB弦与CD弦的交点*)
m=Simplify[((a+b)/2)];(OverBar[m])=Simplify[(((OverBar[a])+(OverBar[b]))/2)]; (* AB弦的中点 *)
WW={z,(OverBar[z])}/.Simplify@Solve[{(z-0)((OverBar[z])-0)==1,((z-m)/((OverBar[z])-(OverBar[m])))==u},{z,(OverBar[z])}]//Flatten;(* D、E的坐标 *)
d=Part[WW,1];(OverBar[d])=Part[WW,2];e=Part[WW,3];(OverBar[e])=Part[WW,4];
WW={z,(OverBar[z])}/.Simplify@Solve[{(z-0)((OverBar[z])-0)==1,((z-m)/((OverBar[z])-(OverBar[m])))==v},{z,(OverBar[z])}]//Flatten;(* C、F的坐标 *)
c=Part[WW,1];(OverBar[c])=Part[WW,2];f=Part[WW,3];(OverBar[f])=Part[WW,4];
p=Simplify@Jd[a,b,c,d];(OverBar[p])=Simplify@(OverBar[Jd])[a,b,c,d]; (* AB与CD的交点P的坐标 *)
q=Simplify@Jd[a,b,e,f];(OverBar[q])=Simplify@(OverBar[Jd])[a,b,e,f]; (* AB与EF的交点Q的坐标 *)
PM=Simplify[((m+ (OverBar[m]))/2)-((p+ (OverBar[p]))/2)];(* PM的长度 *) QM=Simplify[((q+ (OverBar[q]))/2)-((m+ (OverBar[m]))/2)];(* QM的长度 *)
Simplify[PM==QM](* PM是否等于QM？*)