proofs of lemmas 1and 2

Lemma 1
If tεA(2) and s1, are real numbers with s1+s2+...+sn = 1, then
for any vectors p1,...,pn, t(Σsipi) = Σsit(pi).

Let p = Σsipi, and let t(x) = Ax+b. Then
t(p) = Ap+b
= A(Σsipi) + Σsib, as Σsi = 1
= Σsi(Api+b)
= Σsit(pi)

Lemma 2
If tεA(2) fixes points P and Q, then it fixes every point of the line PQ.

Let P and Q have position vectors p and q respectively. Then any R on PQ
has position vector r = sp+(1-s)q, for some real s. Since s+(1-s) = 1, we
can apply Lemma 1 to get t(r) = st(p)+(1-s)t(q) = sp+(1-s)q = r, since t
fixes P and Q. Thus t fixes R.

fixed point theorems page