Altered from Another Problem

Geometry Level 3

Let \(ABC\) be a scalene triangle and \(M\) be the midpoint of \(BC\). The incircle centered at \(I\) touches \(BC\) at \(D\). Denote by \(N\) the midpoint of \(AD\).

Transpose the figure to the complex plane and find \((N-I)(\overline{M} - \overline{I}) - (\overline{N} - \overline{I})(M-I)\) , where N, M, I denote the complex numbers of their respective points.

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