Let "a", "b" and "c" be the sides of the triangle in a way that the side "a" is opposite to the angle of 60°, 
the side "b" is opposite to the angle 50° and the side "c" is opposite to the angle 70°.

We can write these three equality for the area of the triangle:

 = ,    (1)

 = ,    (2)

 = .    (3)

Now, multiply all three equality (1), (2) and (3) (both sides). You will get

 = .

It gives 

 = ,

or, after taking the square root from both sides,

 = .    (4)

Next, let us rewrite the equalities (1) - (3) in the form

 = ,          (1')

 = ,          (2')

 = .          (3')

We are just at the finish line. Divide (4) by (1'). You will get

c =  = .      (5)

Divide (4) by (2'). You will get

a =  = .      (6)

Finally, divide (4) by (3'). You will get

b =  = .      (7)

Formulas (5), (6) and (7) are the solution of the problem. They allow calculate the sides.
If you want to get numerical values, use  sin(50°) = 0.766044,  sin(60°) = 0.866025  and  sin(70°) = 0.939693. From (5), (6) and (7) you will get

a = 15.5117,  b = 13.7209  and  c = 16.8312  (approximately).