2019 IMO Shortlist G8

Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\right), f\left(\ell_{3}\right)$ and $X$ lie on a circle. Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$. (Australia)

令 $\mathcal{L}$ 为平面中所有直线的集合，并令 $f$ 为函数，为 $\ell \in \mathcal{L}$ 中的每条直线分配 $\ell$ 上的点 $f(\ell)$。假设对于任意点$X$，以及通过$X$的任意三条直线$\ell_{1}、\ell_{2}、\ell_{3}$，点$f\left(\ell_{1}\right)、f\left(\ell_{2}\right)、f\left(\ell_{3}\right)$和$X$位于圆上。证明对于任何穿过$P$的线$\ell$，存在唯一点$P$使得$f(\ell)=P$。 (澳大利亚)