2019 IMO Shortlist G2

Let $A B C$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $B C, C A$, and $A B$, respectively. Denote by $\omega_{B}$ and $\omega_{C}$ the incircles of triangles $B D F$ and $C D E$, and let these circles be tangent to segments $D F$ and $D E$ at $M$ and $N$, respectively. Let line $M N$ meet circles $\omega_{B}$ and $\omega_{C}$ again at $P \neq M$ and $Q \neq N$, respectively. Prove that $M P=N Q$. (Vietnam)

设$A B C$ 为锐角三角形，$D、E$ 和$F$ 分别为从$A、B$ 和$C$ 到边$B C、C A$ 和$A B$ 的高度的英尺。用$\omega_{B}$和$\omega_{C}$表示三角形$B D F$和$C D E$的内切圆，并让这些圆分别在$M$和$N$处与线段$D F$和$D E$相切。让$M N$ 线再次分别在$P \neq M$ 和$Q \neq N$ 处与圆$\omega_{B}$ 和$\omega_{C}$ 相交。证明$M P=N Q$。 (越南)