Since

[math]AB : BC : CD = 1 : 2 : 3[/math]

then along the whole side,

[math]AB = 1[/math]

[math]AC = 1+2= 3[/math]

[math]AD = 1+2+3 = 6[/math]

So the corresponding side lengths of the similar triangles are:

[math]ABG : ACF = 1 : 3[/math]

[math]ABG : ADE = 1 : 6[/math]

Using the fact that areas of similar triangles are proportional to the square of corresponding sides:

[math]\text{area of } ABG : \text{area of } ACF= 1^2 : 3^2 = 1 : 9[/math]

and

[math]\text{area of } ABG : \text{area of } ADE = 1^2 : 6^2 = 1 : 36[/math]

Therefore:

[math]\text{area of } ABG = 1[/math]

[math]\text{area of } ACF = 9[/math]

[math]\text{area of } ADE = 36[/math]

So:

[math]\text{area of } BCFG = Triangle ACF – Triangle ABG = 9-1 = 8[/math]

and

[math]\text{area of } CDEF = Triangle ADE – Triangle ACF = 36-9 = 27[/math]

Therefore,

[math]\text{area of } ABG : \text{area of } BCFG : \text{area of } CDEF = 1 : 8 : 27[/math]