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Five solutions to triangle problems are compulsory in senior one mathematics.
(1), using sine theorem, Sina/a = sinb/b = sinc/c = m; We know that a+b=√2c,

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