Team selection tests for BMO 2018

Let $ABC$ be a triangle with $M,N,P$ as midpoints of the segments $BC,CA,AB$ respectively. Suppose that $I$ is the intersection of angle bisectors of $\angle BPM$, $\angle MNP$ and $J$ is the intersection of angle bisectors of $\angle CNM$, $\angle MPN$. Denote (${\omega }_1$) as the circle of center $I$ and tangent to $MP$ at $D$, $\left({\omega }_2\right)$ as the circle of center $J$ and tangent to $MN$ at $E$.

1. Prove that $DE$ is parallel to $BC$.

2. Prove that the radical axis of two circles (${\omega }_1$), (${\omega }_2$) bisects the segment $DE$.