三道几何题

tm__xk

1.向量
AM垂直EF <==> (AB+AF)EF=0 <==> AB*2DG-AB*DF+AF*2DG=0
<==> AD*DG+BD*DF+(AD*DG+DF*DG)=0 <==> 2AD*DG+BG*DF=0
<==> 2|AD||DG|=|BG||DF|cosBFD <==> 2sinDBF=sinCDFcosBFD
记DBF=x,BFD=y,往证(cos(x+y)=sinx/siny ==> 2sinx=sin(x+y)cosy).
事实上,
cos(x+y)=sinx/siny ==> 2sinx=2sinycos(x+y)=sin(x+2y)-sinx ==> 4sinx=sinx+sin(x+2y)=2sin(x+y)cosy

2.A处切线和AB,AC,AG构成调和线束.

3.AC上的内切圆切点,旁切圆切点,BF与内切圆的交点组成三角形,EO为中位线,故BF过AC上内切圆切点的对径点.