Graphics Reference
In-Depth Information
Figure 2.8.
Moving two points to another two
points.
T(P
2
)
P
2
a
P
1
¢
P
2
¢
P
1
C
B
D
A
X = R(C)
Figure 2.9.
Proving Lemma 2.2.5.1.
Let M and M¢ be the motions defined in Case 2 that send
P
1
and
P
2
to
P
1
¢ and
P
2
¢, respectively. By hypothesis, |
P
i
¢M(
P
3
)| = |
P
i
¢
P
3
¢| for i = 1,2. The next lemma
shows that either M or M¢
does what we want, namely, either M(
P
3
) =
P
3
¢
or
M¢(
P
3
) =
P
3
¢.
2.2.5.1. Lemma.
Let
A
,
B
, and
C
be three noncollinear points in the plane. The
only vectors
X
in the plane that satisfy the two equations |
AX
| = |
AC
| and |
BX
| = |
BC
|
are
X
=
C
and
X
= R(
C
), where R is the reflection about the line
L
determined by
A
and
B
.
Proof.
Assume that
X
π
C
.
1
2
Claim.
The midpoint
D
=
(
C
+
X
) of the segment [
C
,
X
] lies on the line
L
.
See Figure 2.9. Once the claim is proved we are done because the definition of
D
implies that
X
=
C
+ 2
CD
, which is where the reflection sends the point
D
. Consider
the following identities:
1
2
1
2
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
(
)
-
(
)
-
CD • AD
=
C
+
X
C
•
C
+
X
A
1
2
1
2
(
)
(
)
=
AX
-
AC
•
AC
+
AX
1
4
0
(
)
2
2
=
AX
-
AC
=
