It is known that a+b+c = (√ 2+1) c; c = 1； a+b =√2； S= 1/2*absinC is ab =1/3;

2abcosc=a^2+b^2-c^2=(a+b)^2-2ab-c^2=2-2/3- 1= 1/3；

cosC = 1/2； C = 60

(2) Using sine theorem, we can get

2R(sinA-sinC)(a+c)=(√2a-b)* b；

Using the equal circumferential angles of chords, 2r * sina = a and 2r * sinc = c are obtained.

a^2-c^2=√2ab-b^2； Available COSC = (√ 2)/2 >; 0; So there is sinc = (√ 2)/2;

c = 2r sinc =√2 * R；

a^2-2r*r=√2ab-b^2； 2ab & lt=a^2+b^2=√2ab+2r^2；

So there is AB.

s = 1/2 * ab * sinC & lt； =(√2+ 1)/2*R^2