The Art and Craft of Problem Solving

8.5 TRANSFORMATIONS 305

90 °. Given just the location of points A , B, e, show how to locate the points X, Y, Z.

Solution: Begin with wishful thinking, and assume that we have located the un­
known points X , Y, Z. In reality, we don't know where to draw any of the line segments
shown in the picture below. But there is some structure to grab onto. The triangles with
vertex angles at the points A, B, e are isosceles. Isosceles triangles are symmetrical:
one can rotate one side so that it coincides with the other side.

A

, ,

, ,

, ,

X

Y

, ,

, ,

C

• 
  
  
  
  • --- -. B
    , ,
    ,
    , ,
    , ,
  
  
  
  

" Z

Now that we are thinking about rotations, we search for fixed points. Denote the
rotation about A by 60 ° counterclockwise (Le., the rotation that takes AX to AY) by
RA. Likewise, define RB and Re. Then RA(X ) = Y, RB(Y ) = Z, and Rc(Z) = X. In
other words, Re 0 RB 0 RA takes X to itself!

By 8.5.7, Re 0 RB 0 RA is a rotation by (^60) + (^120) + 90 = 270 °. Rotations have ex­
actly one fixed point, namely, their center, so X is the center of Re 0 RB 0 RA. Two
applications of 8.5 .8 locate X. Once we find X, we can easily locate the other two
points, since RA(X ) = Y and RB(Y ) = Z. •

Homothety

Many important transformations are not rigid motions. For example, a similarity is
any transformation that preserves ratios of distances. The simplest of these are the
homotheties, also called central similarities. A homothety with center e and ratio

k maps each point X to a point X' on the ray ex so that ex' /ex = k. Here is an

example, with center at e and k = 1/2.

A' _f/j---

c:<::::KfJ,

B' ---___

_

B