如图,在四边形ABCD中,AC与BD相交于点O,直线l平行于BD,且与AB、DC、BC、AD及AC的延长线分别相交于点M、N、R、S和P,求证:PM•PN=PR•PS.
问题描述:
如图,在四边形ABCD中,AC与BD相交于点O,直线l平行于BD,且与AB、DC、BC、AD及AC的延长线分别相交于点M、N、R、S和P,求证:PM•PN=PR•PS.
答
证明:∵直线l平行于BD,
∴
=PN OD
=CP CO
,得PR OB
=PN PR
①,OD OB
=PM OB
=AP AO
,得PS OD
=PS PM
②,OD OB
由①②得
=PN PR
,即PM•PN=PR•PS.PS PM