Could you derive something useful from the data?
We have before us an unsolved problem, an open question.
We have to find the connection between the data
and the unknown.  We may represent our unsolved problem
as open space between the data and the unknown,
as a gap across which we have to construct a bridge.  We
can start constructing our bridge from either side, from
the unknown or from the data.
   Look at the unknown! And try to think of a familiar 
problem having the same or similar unknown. This
suggests starting the work from the unknown.
      Look at the data! Could you derive something useful
from the data? This suggests starting the work from the
data. 
     It appears that starting the reasoning from the unknown
is usually preferable (see PAPPUS and WORKING BACKWARDS.)
Yet the alternative start, from the data, also
has chances of success, must often be tried, and deserves
illustration.
      Example.  We are given three points A,B, and C.  Draw
a line through A which passes between B and C and is
at equal distances from B and C.
   What are the data Three points, A, B, and C.
are given in position. we Draw a figure,exhibiting the data
(Fig 13.)
     What is the unknown? a straight line.
     What is the condition? The required line passes 
through A, and passes between B and C, and the same
distance from each. We assemble the unknown and the data


                   B 
                    .


     
         A .

                         .
                         C

                Fig. 13

In a figure exhibiting the required relations (fig 14).
Our figure, suggested by the Definition of the distance of
a point from a straight line, shows the right angles 
involved by this definition.

                   B 
                    *
                    |
                    |
                    |
         A *--------------------
                         |
                         *
                         C
               Fig. 14

The figure, as it is plotted, is still "too empty." The
unknown straight line is still unsatisfactory connected
with the data A, B, and C.  The figure needs some auxiliary
line, some addition--but what? A fairly good student
can get stranded here.  There are, of course, various 
things to try, but the best question to refloat him is:
Could you derive something useful from the data?
    In fact, what are the data? The three points exhibited
in fig 13, nothing else.  We have not yet used sufficiently 
the points B and C; we have to derive something useful
from them. But what can you do with just two points?
    Join them by a straight line.  So, we draw Fig. 15.


                   B 
                    .
                     \
                      \
                       \
         A .            \
                         \
                          .
                          C

              Fig 15

   If we suppose Fig. 14 and Fig. 15, the solution may
appear in a flash: There are two right triangles, there are 
congruent, there is an all-important new point of inter
section.