过抛物线y^2=2px(p>0)的焦点F的直线与抛物线相交于M,N两点,自M,N向准线l做垂线,垂足分别为M1,N1

S2^2=4S1S3成立,证明如下：
证：设M（x1,y1）,N（x2,y2）
则由抛物线的定义得
|MM1|=|MF|= x1+p/2,|NN1|=|NF|= x2+p/2,
于是
S1= 1/2|MM1||F1M1|= 1/2(x1+p/2)|y1|,
S2= 1/2|M1N2||FF1|= 1/2p|y1-y2|,
S3= 1/2|NN1||F1N1|= 1/2(x2+p/2)|y2|,
∵S2^2=4S1S3 互推 (1/2p|y1-y2|^2=4×1/2(x1+p/2)|y1|• 1/2(x2+p/2)|y2|
互推 [1/4p^2(y1+y2)^2-4y1y2]= [x1x2+p/2(x1+x2)+p^2/4]|y1y2|,
将 {x1=my1+p/2 x2=my2+p/2与 {x1+y2=2mp y1y2=-p^2代入上式化简可得
p^2（m^2p^2+p^2）=p^2（m^2p^2+p^2）,此式恒成立．
故S2^2=4S1S3成立．