Take $A=(a,a^2), C=(c,c^2)$ $M=\left(\frac{a+c}{2},\frac{a^2+c^2}{2}\right)$ $B,D$ are the only two points $(x,y)$ such that $y=x^2\wedge \left(x-\frac{a+c}{2}\right)^2+\left(y-\frac{a^2+c^2}{2}\right)^2=|AC|^2/4=\left(\frac{a-c}{2}\right)^2+\left(\frac{a^2-c^2}{2}\right)^2$ which gives $(x-a)(x-c)(x^2+(a+c)x+ac+1)=0$ which obviously means that $x^2+(a+c)x+ac+1=0$ As $b,d$ are roots of this equation, then $b+d=-(a+c)\wedge bd=ac+1\implies b^2+d^2=a^2+c^2-1$ But $N=\left(\frac{b+d}{2},\frac{b^2+d^2}{2}\right)$ so projection of the segment $MN$ on the axis $OY$ equals to $|\frac{b^2+d^2}{2}-\frac{a^2+c^2}{2}|=\frac{1}{2}$