306 CHAPTER^8 GEOMETRY FOR AMERICANS
Homotheties are a transfonnational way of thinking about similarity. Not all similari
ties are homotheties, but all homotheties are similarities.
As with other transformations, it is crucial to think about invariants. Clearly,
they have just one fixed point-the center-but homotheties leave other things un
changed.
Fact 8.5.11 Homotheties map lines to parallel lines. Consequently, homotheties pre
serve angles.? Conversely, if two figures are directly similar, non congruent, and cor
responding sides are parallel, then they are homothetic.^8
Fact 8.5 .11 gives us a nice criterion for concurrence. If two figures are directly
similar, with corresponding sides parallel, the lines joining corresponding points con
cur, namely at the center of the homothety.
Whenever a problem involves midpoints, look for a homothety with ratio 1/2.
Let's apply this idea to get a new solution to a problem that we solved in Exam
ple 4.2. 14 using complex numbers.
Example 8.5.12 (Putnam 19 96) Let CI and C 2 be circles whose centers are 10 units
apart, and whose radii are 1 and 3. Find, with proof, the locus of all points M for which
there exist points X on C I and Y on C 2 such that M is the midpoint of the line segment
XY.Solution: LetA, B denote the centers ofCI andC 2 , respectively. For the time being, fix
X on C I and let Y move along C 2. Notice that M is the image of Y under a homothety
with center X and ratio 1/2. Thus, as Y moves about C 2 , the point M traces out a circle
whose radius is 3/2. Where will this circle be located? Its center will be the image of
B under this homothety, in other words, the midpoint of the line BX.
Now that we know what happens when X is fixed and Y moves freely around CI,
we can let X move freely around C I. Now the locus of points M will be a set of circles,
each with radius 3/2. What is the locus of their centers? Each center is the midpoint(^7) For those of you who understand directed length, we are assuming that the ratio k is positive here.
(^8) Two figures are called directly similar, as opposed to oppositely similar, if corresponding angles are equal
in magnitude as well as orientation. In other words, if you can tum one figure into the other by translating,
rotating, and then shrinking or magnifying, they are directly similar. If you have to lift the figure off the page and
tum it over, or equivalently, perform a reflection, then they are oppositely similar.