Sec. 10.11 A case of Pascal’s theorem, 1640 181
thenP,Q,R,Sare the vertices of a parallelogram.
10.2 IfZ 1 ∼z 1 ,Z 2 ∼z 2 andZ 3 ∼z 3 are non-collinear points show that
z 1 (z ̄ 3 −z ̄ 2 )+z 2 (z ̄ 1 −z ̄ 3 )+z 3 (z ̄ 2 −z ̄ 1 )= 4 iδF(Z 1 ,Z 2 ,Z 3 )= 0.
10.3 LetA∼a,B∼b,C∼cbe non-collinear points andP∼pa point such that
AP,BP,CPmeetBC,CA,ABatD∼d,E∼e,F∼f, respectively. Show for
sensed ratio that
BD
DC
=
p(b ̄−a ̄)+p ̄(a−b)+ba ̄−ab ̄
p(a ̄−c ̄)+p ̄(c−a)+ac ̄−ca ̄
,
and hence prove Ceva’s theorem that
BD
DC
CE
EA
AF
FB
= 1.
10.4 LetA∼a,B∼b,C∼cbe non-collinear points. Given any pointP∼p,show
that as(c−a)/(b−a)is non-real there exist unique real numbersyandzsuch
thatp−a=y(b−a)+z(c−a),andsop=xa+yb+zcwherex+y+z=1.
Show that ifAPmeetsBCit is in a pointD∼dsuch that
d=
1
1 +r
b+
r
1 +r
c,
wherer=z/y. Hence prove Ceva’s theorem that ifD∈BC,E∈CA,F∈AB
are such thatAD,BE,CFare concurrent, then
BD
DC
CE
EA
AF
FB
= 1.
10.5 LetA∼a,B∼b,C∼cbe non-collinear points. IfD∼d,E∼ewhere
d=
1
1 +λ
b+
λ
1 +λ
c,e=
1
1 +μ
c+
μ
1 +μ
a,
andDEmeetsABit is in a pointF∼fwhere
f=
1
1 +ν
a+
ν
1 +ν
b,
andλμν=−1.
Prove Menelaus’ theorem that ifD∈BC,E∈CA,F∈ABare collinear, then
BD
DC