蝴 蝶 定 理 复 数 法 证 明 1

M 是 AB 弦的中点，求证 PM=QM

Clear["Global`*"];(* 令 ABCDEF 都在单位圆上，设D、F为自由点，各点坐标用 a,b,d,f 表达 *)
(OverBar[a])=1/a; (OverBar[b])=1/b; (OverBar[d])=1/d;  (OverBar[f])=1/f; 
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-d)/((OverBar[z])-(OverBar[d])))==((m-d)/((OverBar[m])-(OverBar[d])))},{z,(OverBar[z])}]//Flatten;(* C点坐标 *)
c=Part[WW,1];(OverBar[c])=Part[WW,2];Print["C = ",c];
WW={z,(OverBar[z])}/.Simplify@Solve[{(z-0)((OverBar[z])-0)==1,((z-f)/((OverBar[z])-(OverBar[f])))==((m-f)/((OverBar[m])-(OverBar[f])))},{z,(OverBar[z])}]//Flatten;(* E点坐标 *)
e=Part[WW,1];(OverBar[e])=Part[WW,2];Print["E = ",e];
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*)
p=Simplify@Jd[a,b,e,d];(OverBar[p])=Simplify@(OverBar[Jd])[a,b,e,d]; (* AB与ED的交点P的坐标 *)
q=Simplify@Jd[a,b,c,f];(OverBar[q])=Simplify@(OverBar[Jd])[a,b,c,f]; (* AB与CF的交点Q的坐标 *)
Print["P = ",p];Print["Q = ",q];
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 *)
Print["PM = ",PM];Print["QM = ",QM];
Print["可见 PM = QM，蝴蝶定理得到证明。 " ];