54 CHAPTER 1 Advanced Euclidean Geometry
5.^11 In triangleABC, pointDis on [BC] withCD= 2 andDB= 5,
pointE is on [AC] withCE = 1 andEA= 3,AB= 8, and [AD]
and [BE] intersect atP. PointsQandRlie on [AB] so that [PQ]
is parallel to [CA] and [PR] is parallel to [CB]. Find the ratio of
the area of 4 PQRto the area of 4 ABC.
6. In 4 ABC, let E be on [AC] with AE : EC = 1 : 2, let F be
on [BC] with BF : FC = 2 : 1, and let G be on [EF] with
EG:GF = 1 : 2. Finally, assume thatDis on [AB] withC, D, G
colinear. FindCG:GDandAD:DB.
7. In 4 ABC, letE be on [AB] such thatAE:EB= 1 : 3, letDbe
on [BC] such thatBD :DC = 2 : 5, and letF be on [ED] such
thatEF :FD = 3 : 4. Finally, let Gbe on [AC] such that the
segment [BG] passes throughF. FindAG:GC andBF :FG.
8. You are given the figure to the right.
(a) Show thatBJ :JF = 3 : 4 and
AJ:JE = 6 : 1.
(b) Show that
DK:KL:LC=
EJ:JK:KA=
FL:LJ :JB= 1 : 3 : 3.
(c) Show that the area of 4 JKLis one-seventh the area of 4 ABC.(Hint: start by assigning masses 1 toA, 4 toBand 2 toC.)- Generalize the above result by replacing “2” byn. Namely, show
that the area ratio
area 4 JKL: area 4 ABC= (n−1)^3 : (n^3 −1).(This is a special case ofRouth’s theorem.)(^11) This is essentially problem #13 on the 2002American Invitational Mathematics Exami-
nation(II).