I've derived equations for 2d polygon's moment of inertia using Green's Theorem (constant density \rho)

$$I_y = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + x_i x_{i+1} + x_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$$

$$I_x = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( y_i^2 + y_i y_{i+1} + y_{i+1}^2 ) ( x_{i+1} y_i - x_i y_{i+1} )$$

And I'm trying to add them up for calculating $I_0 = I_x + I_y$.

$$I_0 = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 - y_i^2 + x_i x_{i+1} - y_i y_{i+1} + x_{i+1}^2 - y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$$

But I found different(?) equation for $I_0$ on the internet. and many people says below equation is correct.

$$I_0 = \frac{\rho}{6} \frac{ \sum_{i=0}^{i=N-1} ( x_i^2 + y_i^2 + x_i x_{i+1} + y_i y_{i+1} + x_{i+1}^2 + y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i ) }{ \sum_{i=0}^{i=N-1} ( x_i y_{i+1} - x_{i+1} y_i ) }$$

So I'm confusing now. I think my equations for $I_x$ and $I_y$ is correct. But how am I gonna calculate for $I_0$ (moment of inertia with respect to origin axis). I couldn't prove both equations are equal.

Could you help me out please ?

(This post has been cross-posted at http://math.stackexchange.com/questions/59470/calculating-moment-of-inertia-in-2d-planar-polygon)