Simplify symbolic expression for the Taylor series

I have written some code that generates a system of symbolic, nonlinear equations based on Taylor series expansions. For example, an equation with order 2 is

eqn1 = y1/2 - x2/6 - x1/2 + y2/6 + (x1 - y1)^2/8 + ((x1/3 - y1/3)*(x2 - y2))/8 + ((x2/3 - y2/3)*(x1 - y1))/8 + ((x2/9 - y2/9)*(x2 - y2))/8

where the x's and the y's are symbolic variables. This expression can obviously be simplified even further into

eqn1_simplified = y1/2 - x2/6 - x1/2 + y2/6 + (x1 - y1)^2/8 + ((x1 - y1)*(x2 - y2))/12 + ((x2 - y2)^2/72

but it seems like the coefficients in the brackets are preventing me from doing so. I have tried using simplify and simplifyfraction but it just returns the expression in eqn1.

[madhan ravi]

how do you want the result look like?

[Hau Kit Yong]

I want eqn1 to be simplified into eqn1_simplified.