312 CHAPTER 8 GEOMETRY FOR AMERICANS
of CD, B' the midpoint of DE, C' the midpoint of EA,
D' the midpoint of AB, and E' the midpoint of BC.
Show that the midpoints of A'C', C' E', E' B', B'D' and
D' A' all lie on a circle.
8.5.22 Solve Example 8.4.2 using translation: trans
late the triangle with parallel motions, to create par
allelograms, and translate the trisection lines as well.
Look for congruent shapes that can be reassembled.
If that seems too daunting, try the following eas
ier problem first: Let ABCD be a rectangle, and let
E, F, G, H be the midpoints of AB,BC,CD,DA, re
spectively. Lines AF,BG,CH,DE intersect pairwise to
form a quadrilateral inside ABCD. Prove that the area
of this quadrilateral is [ABCD]/5.
8.5.23 (Hungary 1940) Let T be an arbitrary triangle.
Define M(T) to be the triangle whose sides have the
lengths of the three medians of T.
(a) Show that M(T) actually exists for any triangle
T. Show how to construct M (T) with compass
and ruler, given T.
(b) Find the ratio of the areas of M(T) and T.
(c) Prove that M(M( T)) is similar to T.
8.5.24 Prove that the lines AB and CD are perpendic
ular if and only if AC^2 -AD^2 = BC^2 - BD^2.
8.5.25 Let ABC be a triangle with a = BC, b =
AC, c = AB. Denote the incenter, circumcenter, and
orthocenter by [, K, and H, respectively.
(a) Show that A + E +C = 2K + ii. (In other words,
if the origin is located at the circumcenter, then
the orthocenter is given by the vector sum of the
vertices.)(b) Sh h [-
aA + bE + cC
ow t at =.
a+b+c
8.5.26 Prove that for any parallelogram ABCD, the
following holds:(^2) (AB^2 +AD^2 ) = AC^2 +BD^2.
8.5.27 Let p, q, r be lines meeting in a single point,
and let Fa denote reflection across line a. Show that
there is an unique line t such that Fp 0 Fq 0 Fr = Ft.
8.5.28 Carefully prove Facts 8.5.7 and 8.5.9. Also,
show how to find the centers of the new rotations pro
duced.
8.5.29 We proved most of the statement, "all rigid
motions are translations, rotations, reflections, or glide
reflections." What was left out of the proof? Supply
the rest.
8.5.30 Given an arbitrary triangle ABC, construct
equilateral triangles on each side (externally), as
shown below.
A
......... ---y
z
Show that AX = BY = CZ.
8.5.31 Using compass and ruler, construct an equilat
eral triangle whose vertices lie on three given concen
tric circles.
8.5.32 Prove that if squares are erected externally on
the sides of a parallelogram, then their centers are the
vertices of a square.
8.5.33 (Canada 1989) Let ABC be a right triangle of
area 1. Let A' , B' , C' be the points obtained by reflect
ing A, B, C, respectively, in their opposite sides. Find
the area of 6.4' B'C'.
8.5.34 Given a triangle ABC, construct (with ruler and
compass) a square with one vertex on AB, one vertex
on AC, and two (adjacent) vertices on BC.
8.5.35 Rotate an arbitrary triangle ABC about its cen
troid M by 1800 to get a six-pointed star as shown be
low.
A
A'
Find the area of this star in terms of [ABC].
8.5.36 The Pantograph. The diagram below depicts
a pantograph, a device used for making scale copies,
still found in engraving shops. The pantograph lies on
a planar surface (possibly a sheet of paper or plastic
or metal), fixed to it at F. Segments F B and BP are
fixed lengths, hinged at B. Likewise, AP' and P'C are
fixed lengths, hinged at P'. Points A and C are also