Sec. 10.11 A case of Pascal’s theorem, 1640 18310.15 If[Z 1 ,Z 2 ,Z 3 ,Z 4 ]is a parallelogram,Wis a point on the diagonal lineZ 1 Z 3 ,a
line throughWparallel toZ 1 Z 2 meetsZ 1 Z 4 andZ 2 Z 3 atW 1 andW 2 respectively,
and a line throughWparallel toZ 1 Z 4 meetsZ 1 Z 2 andZ 3 Z 4 atW 3 andW 4 ,
respectively, prove that
δF(W,W 4 ,W 1 )=δF(W,W 3 ,W 2 ).10.16 IfZ 1 =Z 2 andδF(Z 1 ,Z 2 ,Z 3 )=−δF(Z 1 ,Z 2 ,Z 4 ), prove that the mid-point of
Z 3 andZ 4 is onZ 1 Z 2.
10.17 Suppose thatZ 1 ,Z 2 ,Z 3 ,Z 4 are points no three of which are collinear. Show that
[Z 1 ,Z 3 ]∩[Z 2 ,Z 4 ]=0 if and only if/
δF(Z 1 ,Z 2 ,Z 4 )
δF(Z 3 ,Z 2 ,Z 4 )<0andδF(Z^2 ,Z^1 ,Z^3 )
δF(Z 4 ,Z 1 ,Z 3 )