(1).AD=√(AC²-DC²)=√(9-1)=2√2.
S△ABC=BC*AD/2=2*2√2/2=2√2.
sin(A/2)=DC/AC=1/3,cos(A/2)=√(1-sin²A/2)=2√2/3.
sinA=2sin(A/2)cos(A/2)=2*(1/3)*(2√2/3)=4√2/9
BE=AB*sinA=3*(4√2/9)=4√2/3
(2).
BC*r/2+2AC*r/2=4√2/3,r+3r=4√2/3,r=√2/3.△ABC的内切圆的半径r=√2/3.
∠BOC=2∠A,sinA=4√2/9,cosA=√(1-32/81)=√(49/81)=7/9
sin(2A)=2sinAcosA=2*(4√2/9)*(7/9)=56√2/81
cos(2A)=√(1-6272/6561)=√(289/6561)=17/81
BC²=R²+R²-2R²cos(2A)
4=2R²(1-17/81)
R²=2/(64/81)=2*81/64
R=9√2/8 △ABC的外接圆的半径R=9√2/8