8.1 THREE "EASY" PROBLEMS 257with the basic facts, we will explore a few more advanced ideas, many of them con
nected by the deep concept of transformations. This idea, pioneered by Felix Klein
and Henri Poincare and others in the 19 th century, revolutionized geometry, and con
nected it with many other branches of mathematics. We will only scratch the surface
of this subject, though.
This is just a single chapter, so we are compelled to omit many topics. Rather
than cover many concepts, we prefer to revisit the classical ones carefully, so that you
can master them from a problem solver's perspective. Consequently, we leave out
fascinating topics such as projective, solid, and hyperbolic geometry. But we will not
hesitate to temporarily replace classical methods with algebra or trigonometry. After
all, problem solving is our goal!
The typical geometry problem consists of two parts. First, a diagram is specified,
usually with words; it is often quite challenging to actually draw the diagram correctly.
This initial stage often frustrates beginners, who do not expend enough energy and
investigation to draw a picture carefully.
The second stage of the problem generally asks to prove a "rigidity" statement
about the diagram; some property or properties that do not change no matter how the
diagram is drawn.
Here are three examples. All are "elementary," but none are really easy to prove.
These three problems are a diagnostic test of your skill in geometrical problem solving.
If you have little difficulty with them, you can safely skim the next section (but do
carefully check your work on pages 275-278, where we show the solutions, and look
at all of the problems at the end of the section). Otherwise, you should read this next
section very slowly and carefully, doing as many problems as you can.
8. 1.1 The Power of a Point Theorem. Given a fixed point P and a fixed circle, draw
a line through P that intersects the circle at X and Y. The power of the point P with
respect to this circle is defined to be the quantity PX. PY.
The Power of a Point Theorem (also known as POP) states that this quantity is
invariant; i.e., does not depend on the line that is drawn. For example, in the picture
below,PX ·PY = PX'·Py'.
Prove POP. (There are three cases to consider, depending on whether P lies on, inside,
or outside the circle.)