1.点P为圆O外一点,PS、PT是两条切线,过点P作圆O的割线PAB,交圆O于A,B两点,与ST交与点C.求证1/PC=1/2(1/PA+1/PB)2.PA是圆O的切线,从PA的中点B作割线BCD,交圆O与C,D两点,连结PC和PD,分别交圆与E和F.求证PA‖

Question

1.点P为圆O外一点,PS、PT是两条切线,过点P作圆O的割线PAB,交圆O于A,B两点,与ST交与点C.求证 1/PC=1/2(1/PA+1/PB)
2.PA是圆O的切线,从PA的中点B作割线BCD,交圆O与C,D两点,连结PC和PD,分别交圆与E和F.求证PA‖EF

Answer 1

1.设PA=X,PB=Y,PC=Z,OS=OT=R
联结OP交ST于D,联结OS
PS^2=PA*PB=XY
OP^2=OS^2+PS^2=XY+R^2
由OP*PD=PS^2得PD=XY/(R^2+XY)^(1/2)
SD^2=SP^2-PD^2=R^2*XY/(R^2+XY)^(1/2)
由SC*CT=AC*BC得SD^2-CD^2=(Z-X)*(Y-Z)
CD^2=R^2*XY/(R^2+XY)-ZY+Z^2+XY-XZ
CD^2+PD^2=PC^2将上面的结果代入
R^2*XY/(R^2+XY)-ZY+Z^2+XY-XZ＋(XY)^2/(R^2+XY)=Z^2
整理得1/Z=1/2X+1/2Y
即1/PC=1/2PA+1/2PB