>>800
外接円の半径をrとすると
　⊿ABG = (1/2)rr sin(2C) = rr sin(C) cos(C),
　⊿ABC = 2rr sin(A) sin(B) sin(C),
題意より
　⊿ABG = (1/8)⊿ABC,
　cos(C) = (1/4)sin(A)sin(B),
また
　A + B + C = 180°
　A = 60°　　　　　　　　　(← 題意)
これを解いて
　A = 60°
　B = arctan(4/√27) = (1/2)arccos(11/43) = 37.589089468975°
　C = arctan(13/√3) = (1/2)(π - arccos(83/86)) = 82.410910531025°
ところで
　∠AGR = 180°- 2C = 15.178178938°
　∠GAR = 90°- B = 52.410910531025°
正弦定理より
⊿AGR = rr cos(B)sin(2C)/{2cos(B+2C-180)}
　= 0.1122092715867 rr
　= 3,
∴　r = 5.1706632668738
　⊿ABC = 28,
　⊿ABG = 7/2,