[G.Polya]
Auxiliary elements. There is much more in our
conception of the problem at the end of our work than was in it as we
started working (PROGRESS AND ACHIEVEMENT 1). As our work progresses, we add new elements to those
originally considered. An element that we introduce in
the hope that it will further the solution is called an
auxiliary element.
1. There are various kinds of auxiliary elements. Solving
a geometric problem, we may introduce new lines
into our figure, auxiliary lines. Solving an algebraic problem,
we may introduce an auxiliary unknown( AUXILIARY PROBLEMS, 1). An
auxiliary theorem is a theorem whose
proof we undertake in the hope of promoting the solution
of our original problem.
2. There are various reasons for introducing auxiliary
elements. We are glad when we have succeeded in recollecting
a problem related to ours and solved before. it is
probable that we can use such a problem but we do not
know yet how to use it. For instance, the problem which
we are trying to solve is a geometric problem, and the
related problem which we have solved before and have
now succeeded in recollecting in a problem about triangles.
Yet there is no triangle in our figure; in order to
make use of the problem recollected we must have a
triangle; there fore, we have to introduce one, by adding
suitable auxiliary lines to our figure. In general, having
recollected a formerly solved related problem and
wishing to use it for our present one, we must often ask;
" Should we introduce some auxiliary element in order to make its
use possible? (The example in section 10 is typical.)
Going back to definitions
We have another opportunity to introduce auxiliary elements.
For instance,
explicating the definition of a circle we should not only
mention its centre and its radius, but we should also
introduce these geometric elements into our figure. With-
out introducing them, we could not make any concrete
use of the definition; stating the definition without
drawing something is mere lip-service.
Trying to use known results and going back to the definitions
are e among the best reasons for introducing auxiliary
elements;
but they are not only the only ones. We may
add auxiliary elements to the conception of our problem
in order to make it fuller. More suggestive, more familiar
although we scarcely know yet explicitly how we shall
be able to use the elements added. We may just feel that
it is a bright idea" to conceive the problem that way
with such and such elements added.
We may have this or that reason for introducing an
auxiliary element but we should have some reason. We
should not introduce auxiliary elements wantonly.
3. Example. Construct a triangle, being given one
angle, the altitude drawn from the vertex of the given
angle, and the perimeter of the triangle.
A
.-T\
.-~. |.\
.-~ ~| \
.-~ h| \
P .-~ | \
.-~ ~~~~~~~~~~~~~~~~~~~
.-~
.-~
A
.-T\
.-~. |.\
.-~ ~| \
.-~ h| \
.-~ | \
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
P
We introduce suitable notation. . Let a denote the given
angle, h the given altitude drawn from the vertex A of a
and p the given perimeter. We draw a figure in which
we easily place a and h. have we used all the data? No.
our figure does not contain the given length p, equal to
the perimeter of the triangle. Therefor we must introduce
p . but how?
we may attempt to introduce p in various ways. The
attempts exhibited in Figs 9 10 appear clumsy. If we try to make clear
to ourselves why they appear so unsatisfactory, we may perceive that it is
for lack of symmetry.
In fact, the triangle has three unknown sides, a,b,c.
we call a, as usual, the side opposite to A; we know that
a+b+c=p
Now, the sides b and c play the same role; they are e interchangeable;
our problem is symmetric with respect to b and c.
But b and c do not play the same role in our
figures 9, 10, placing the length p we treated b and c
differently, the figures g and 10 spoil the natural symmetry
of the problem with respect to b and c. We should
place p so that it has the same relation to b as to c.
This consideration may be helpful in suggesting to
place, the length p as in fig 11. We add to the side a of
A
.
. /| .
. / | .
. b/ h|| c .
. / | | .
._________/____L__|__________.
E b C a B c D
the triangle the segment CE of length b on one side and
the segment BD of the length c. on the other side so that
p appears in fig 11 as the line ED of length
b + a + c = p
If we have some little experience in solving problems of
construction, we shall not fail to introduce into the
figure, along with ED, the auxiliary lines AD and AE,
each of which is the base of an isosceles triangle. In fact,
it is not unreasonable to introduce elements into he
problem which are particularly simple and familiar, as
isosceles triangle.
We have been quite lucky in introducing our auxiliary
lines. Examining the new figure we may discover that
/_EAD has a simple relation to the given angle a. In fact,
we find using the isosceles triangles /\ABD and /\ACE
that /_DAE = a/2 + 90~, After this remark, it is natural to
try the construction of /\DAE. Trying this construction,
we introduce an auxiliary problem which is much easier
than the original problem.
4. Teacher and authors of textbooks should snot forget
that the intelligent student and THE INTELLIGENT READER
are not satisfied by verifying that the steps of a
reasoning are correct but also and to know the motive and the
purpose of the various steps. The introduction of an
auxiliary element is a conspicuous step. If a tricky
auxiliary line appears abruptly in the figure, without any
motivation, and solves the problem surprisingly, intelligent
students and readers are disappointed; they feel that
they are cheated. Mathematics is interesting in so far as
it occupies our reasoning and inventive powers. But there
is nothing to learn about reasoning and invention if the
motive and purpose of the most conspicuous step remains
incomprehensible. To make such steps comprehensible
by suitable remarks (as in the foregoing under 3) or by
carefully chosen questions and suggestions (as in sections 10,18,19,20)
takes a lot of time and effort; but it may
be worth while.