Lemma 1
If tεA(2) and s_{1},s_{2}...s_{n} are real numbers with
s_{1}+s_{2}+...+s_{n} = 1, then
for any vectors p_{1},...,p_{n},
t(Σs_{i}p_{i}) = Σs_{i}t(p_{i}).
Proof
Let p = Σs_{i}p_{i}, and let t(x) = Ax+b. Then
t(p) 
= Ap+b 


= A(Σs_{i}p_{i}) + Σs_{i}b, 
as Σs_{i} = 1 

= Σs_{i}(Ap_{i}+b) 


= Σs_{i}t(p_{i}) 

Lemma 2
If tεA(2) fixes points P and Q, then it fixes every point of the line PQ.
Proof
Let P and Q have position vectors p and q respectively. Then any R on PQ
has position vector r = sp+(1s)q, for some real s. Since s+(1s) = 1, we
can apply Lemma 1 to get t(r) = st(p)+(1s)t(q) = sp+(1s)q = r,
since t
fixes P and Q. Thus t fixes R.

