2018 IMO Shortlist G7

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $A B C$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A, B, C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $A O P, B O P$, and $C O P$ by $O_{A}, O_{B}$, and $O_{C}$, respectively. The lines $\ell_{A}, \ell_{B}$, and $\ell_{C}$ perpendicular to $B C, C A$, and $A B$ pass through $O_{A}, O_{B}$, and $O_{C}$, respectively. Prove that the circumcircle of the triangle formed by $\ell_{A}, \ell_{B}$, and $\ell_{C}$ is tangent to the line $O P$. (Russia)

设$O$为外心，$\Omega$为锐角三角形$A B C$的外接圆。令$P$ 为$\Omega$ 上的任意点，不同于$A、B、C$ 及其在$\Omega$ 中的对映点。分别用 $O_{A}、O_{B}$ 和 $O_{C}$ 表示三角形 $A O P、B O P$ 和 $C O P$ 的外心。垂直于$B C、C A$ 和$A B$ 的线$\ell_{A}、\ell_{B}$ 和$\ell_{C}$ 分别穿过$O_{A}、O_{B}$ 和$O_{C}$。证明$\ell_{A}、\ell_{B}$和$\ell_{C}$构成的三角形的外接圆与线$O P$相切。 (俄罗斯)