Geometry problem on angle bisectors and intersecting line segments Two equal line segments $AB$ and $CD$ intersect each other at a point $M$. If the perpendicular bisectors of $AD$ and $BC$ intersect each other at the point $N$, prove that the two angles $\angle AMN$ and $\angle CMN$ are equal.
According to the following picture, here is the summary of the problem

Theorem. If the following assumptions hold
  $$\begin{align}
AB&=CD \\
NP &\bot AD \\
AP &= PD \\
NQ &\bot BC \\
BQ &=QC \\
\end{align}$$
  then
  $$\angle AMN = \angle CMN$$


My Attempt:
We know that the distance of every point on the perpendicular bisector from two ends of segment is equal. From that, we can know $BN = CN$ and $AN = DN$. But how can we now prove that $MN$ is the bisector of $\angle AMC$?


Since same colored lines are equal in pairs, $\triangle DNC$ is congruent to $\triangle ANB$.
Result-1 $\gamma$ will be equal to $\delta$. From which, we get $\angle DAN = \angle BCN$.
Result-2 $\alpha = \beta$. This means $ADMN$ is cyclic and therefore $x = \angle DAN$.
Hope you can finish the missing detail.
