2022 IMO Shortlist G7

Let $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ be two triangles having the same circumcircle $\omega$, and the same orthocentre $H$. Let $\Omega$ be the circumcircle of the triangle determined by the lines $A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$. Prove that $H$, the centre of $\omega$, and the centre of $\Omega$ are collinear. (Denmark)

设$A B C$ 和$A^{\prime} B^{\prime} C^{\prime}$ 是具有相同外接圆$\omega$ 和相同垂心$H$ 的两个三角形。令 $\Omega$ 为由线 $A A^{\prime}、B B^{\prime}$ 和 $C C^{\prime}$ 确定的三角形的外接圆。证明$H$、$\omega$的中心和$\Omega$的中心共线。 (丹麦)