1.向量AB=AC-BC=(1,-1),
∴|AB|=√2.
2.|AC|=√(2+2cosα),
AB*AC=1+cosα-sinα,
cosA=AB*AC/(|AB||AC|)=(1+cosα-sinα)/[2√（1+cosα)],
(cosA)^2=(1+cosα-sinα-sinαcosα)/[2(1+cosα)]
=(1-sinα)/2,
sinA=√[(1+sinα)/2],
S=(1/2)|AB||AC|sinA
=√[(1+cosα)(1+sinα)/2],α∈(0,π/2),
设t=sinα+cosα,则sinαcosα=(t^2-1)/2,t∈（1,√2],
S=√{[1+t+(t^2-1)/2]/2}
=(t+1)/2,
当t=√2时,S取最大值(√2+1）/2.