Question 51
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 $DE$.
1. Prove that $DE$ is parallel to $BC$.
2. Prove that the radical axis of two circles ($\omega_{1}$), ($\omega_{2}$) bisects the segment $DE$.