I dont know if this is the smartest way to go about it.

 

Let the sides be a-d,a, a+d.

 

Then the ratio in the required format is 1-a/d:a:1+a/d.

 

Hence, our task is to find a/d

 

a-d = k sinA; a = k sinB; a+d = k sinc

 

Since, sinA, sinB, sinC are also in AP

2sinB = sinA+sinC

 

which gives 2cos(A+C/2) = cos/2

 

Now a/d = sinA+sinC/(sinC-sinA) = tan(A+C/2) cot/2

 

Since  cos(A+C/2) = 0.5*cos/2, tan(A+C/2) = (4-cos²/2)/cos(/2)

Hence a/d = (4-cos²/2)/cos(/2) * cot/2 = (4-cos²/2) /sin(/2)