Elementary Functionsy(^0) x
Fig. 5.3
and
arg(wa -w1)-arg(wa -w2) = arg(za - z1)-arg(z 3 - z 2 )
(see Fig 5.3)
217
The converse proposition also holds, as shown in the following theorem.
Theorem 5.1 Every direct similitude of plane triangles can be obtained
by a linear transformation w = az + b.
Proof Let the vertices w 1 and w 2 correspond to the vertices z 1 and z 2 ( z 1 =f.
z2 ), respectively, and suppose that the triangle with vertices w 1 , w 2 , w is
directly similar to the triangle with vertices zi, z 2 , z. Then we must have
or
w-w 1
w-w 2
W1 - Wz Z1W2 - ZzWI
W= z+ -----
Z1 - Zz Z1 - Zz
which is a linear function.
Theorem 5.2 The set S of all linear transformations forms a group under
composition.
Proof Let T 1 z = a 1 z + b 1 and T 2 z = a 2 z + b 2. We have
T1T2z = T1(a2z + b2) = aia2z + aib2 + b1
which is in S. Similarly, T2T 1 z E S.
The identity in Sis the identity transformation Iz = z, and the inverse
of Tz = az + b is r-^1 z = (1/ a )z - b/ a. Composition of functions is always
associative, so that (T1T2)Ta = T1(T2Ta).