The Art and Craft of Problem Solving

8.2 SURVIVAL GEOMETRY I 263

The next example presents a simple result that can easily be generalized, using
powerful methods of similar triangle analysis. Nevertheless, it is very instructive to see
just how much we can do by manipulating the concepts of angles, parallel lines, and
congruent triangles. Although our proof is rather simple, it features a subtle strategy
that you should master, the phantom point method.

Example 8.2.12 Let ABC be an arbitrary triangle, and let D, E, F be the midpoints

of sides BC,AC,AB, respectively. Lines DE, EF, FD dissect the original triangle into

four congruent triangles.

Solution: Avoid the temptation of similar triangle facts (for example, Fact 8 .3.8

on page 274). They aren't necessary.

We could join midpoints and try to get angle equalities, but we have no tools (at

present, without using similar triangles) that allow us to analyze angles by joining two

midpoints. Instead, let's draw lines that do yield angular information. Start at the

midpoint F, and draw a line through F that is parallel to Be. We know in our heart

that this line will intersect AC at E, but we haven't proven it yet. So, for now, call this

intersection point E'. Likewise, draw another line through F, parallel to AC, meeting

BC at D'. Since BF = FA, that suggests we try to show that 6F D' B � 6AE' F.

A

c

Since FE' II BC, L.AF E' = L.B and FE' A = L.e. Likewise, F D' II AC implies L.F D' B =

L.C. Thus L.F E' A = L.BD' F, which allows us to use the AAS congruence condition to

conclude that 6FD'B � 6AE'F. Thus BD' = FE' and D'F = E'A.

Since E'FD'C is a parallelogram , CE' = D'F and FE' = D'C. Combining these

two equalities with the other pair of equalities yields BD' = D'C and CE' = E' A. In

other words, D' and E' coincide, respectively, with D and E.

Now we can conclude that the line joining the midpoints F and E is parallel to

BC, since E = E'. And now our proof can proceed smoothly, since the same argument

tells us that F D II AC and ED II AB. Repeating the first argument yields

6AEF � 6F DB � 6ECD,

and SSS shows that 6DEF � F BD. •

The "phantom" points D' and E' were ugly constructions, but absolutely nec­

essary. Their purpose is similar to that of the "contrary assumption" in a proof by

contradiction, namely something concrete that one can work with. And like contrary

assumptions, phantom points bow out once their work is done.