The Three Reflections Theorem. The purpose of this
exercise is to prove the following: Suppose are two congruent triangles in the plane.
Then with at most three reflections the points A,B,C can be
moved to the points D,E,F, respectively. We will prove it in
several stages.
Remember that when a point P is reflected across a line ,
we find its image by first constructing a line m through P perpendicular to
, then finding the point Q of intersection of the two lines,
and finally finding the point R such that QR=QP and RP=2QP.
In proving the following results, you may use the standard results in
geometry about congruent triangles, perpendicular lines, isosceles
triangles, etc. If you
have any question about which theorems you may cite, just ask me.
- Suppose A and D are two different points in the plane.
Prove that there is
a line p such that the reflection of D across p is A.
- Suppose that A, B and E are three different points in the
plane such that AB=AE. Prove that there is a line q such that
the reflection of A across q is A, but the reflection of E
across q is B.
- Suppose that A, B, C and F are four different points in
the plane such that AC=AF and BC=BF. Prove that there is a line
r such that the reflection of A across r is A, the reflection
of B across r is B, but the reflection of F across r is C.
- Use the preceding three results to show that if are any two congruent triangles in the plane, then
with at most three reflections the points A,B,C can be
moved to the points D,E,F, respectively. Also, why might you sometimes
need fewer than 3 reflections?
- Make a good conjecture (you do not have to prove it) about a
result analogous to the three reflections theorem, involving two
congruent tetrahedra in three-dimensional space.
- Extra Credit: Make a good illustration of the proof of
the Three Reflections Theorem using
Geometer's Sketchpad.