hinges, with fixed locations along FB and BP, respec
tively, so that FA = Api and AB = piC = CPo At P and
pi, pens (or etching tools) point down onto the surface.
The pantograph operator draws something on the sur
face by moving point P, and pi automatically moves
as well. What does pi draw, and why? Explain how
you can adjust this device to copy at different scales.
h
F P' P8.5.37 (USA Team Selection Test 20(0) LetABCD be
a cyclic quadrilateral and let E and F be the feet of per
pendiculars from the intersection of diagonals AC and
BD to AB and CD, respectively. Prove that EF is per
pendicular to the line through the midpoints of AD and
Be.
8.5.38 Example 8 .5.10 on page 304 can be general
ized in many ways. Try these.
(a) Let the given isosceles vertex angles be arbi
trary measures a, f3, y.
(b) What happens if you are given n points, each
the vertex of an isosceles triangle with vertex
angles ai, a 2 , ... , an. and we must locate the
vertices of an n-gon?
(c) What happens if the sum of the angles is 360°?
Can we use the same argument as before?
(d) What happens when the angles are all equal to
180°?
8.5.39 (Putnam 2004) Let n be a positive integer, n �
2, and put 8 = 2 tt/n. Define points Pk = (k,O) in the
xy-plane. for k = 1, 2 , ... , n. Let Rk be the map that ro
tates the plane counterclockwise by the angle 8 about
the point Pk. Let R denote the map obtained by apply
ing. in order. RI, then R 2 , ... , then Rn. For an arbi
trary point (x,y), find, and simplify, the coordinates of
R(x,y).
8.5.40 Use Fact 8.5.11 to show that the composition
of two homotheties is another homothety. Can you find
the center of the composition?
8.5.41 (Bulgaria 2001) Points Al ,BI ,CI are chosen
on the sides BC, CA, and AB of triangle ABC. Point G
is the centroid of !:::ABC, and Go, Gb, Gc are the cen
troids of !:::ABICI,6BAICI,6CAIBI, respectively.
8.5 TRANSFORMATIONS 313The centroids of !:::AIBICI and 6GaGbGc are denoted
by GI and G 2 , respectively. Prove that G, GI, G 2 are
collinear.
8.5.42 (Hungary 1941) Hexagon ABCDEF is in
scribed in a circle. The sides AB, CD, and EF are all
equal in length to the radius. Prove that the midpoints
of the other three sides are the vertices of an equilat
eral triangle. (There are many possible solutions to this
problem. You may want to also try complex numbers.
if you have mastered the material in Section 4.2.)Inversion Problems
We are putting many inversion problems together, as
a hint for you to try inversion on them. However, you
may want to experiment with inversion on other prob
lems. You may discover interesting alternative solu
tions, and at the very least, you will have fun practicing
this sophisticated skill.
8.5.43 Find a Euclidean construction method for in
verting a point X with respect to a circle ro. as sug
gested by the diagram on page 307.
8.5.44 Complete the proof that inversion takes "cir
cles" to "circles" by examining the following cases (at
the very least, draw careful pictures to convince your
self). Let the inversion be with respect to the circle
ro with center O. Find (and prove that your picture is
correct) the image of:
(a) A line that is tangent to ro.
(b) A line that intersects ro in two points. not pass-
ing through O.
(c) A line that passes through O.
(d) A line that does not intersect ro.
(e) A circle exterior to ro.
(f) A circle interior to ro, passing through O.
(g) A circle intersecting ro in two points, not pass
ing through O.
(h) A circle tangent to ro (there are several sub-
cases here!).
8.5.45 Can you design a "machine," similar to the
pantograph (Problem 8.5.36), that draws the inverse of
a point P as the operator draws P? This machine would
have the useful engineering property of converting cir
cular to linear motion, and vice versa.
8.5.46 Using compass and ruler, draw a circle passing
through a given point P tangent to two given circles.