2008 IMO Shortlist G1

In an acute-angled triangle $A B C$, point $H$ is the orthocentre and $A_{0}, B_{0}, C_{0}$ are the midpoints of the sides $B C, C A, A B$, respectively. Consider three circles passing through $H$ : $\omega_{a}$ around $A_{0}, \omega_{b}$ around $B_{0}$ and $\omega_{c}$ around $C_{0}$. The circle $\omega_{a}$ intersects the line $B C$ at $A_{1}$ and $A_{2} ; \omega_{b}$ intersects $C A$ at $B_{1}$ and $B_{2} ; \omega_{c}$ intersects $A B$ at $C_{1}$ and $C_{2}$. Show that the points $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ lie on a circle.

在锐角三角形 $A B C$ 中，点 $H$ 为垂心，$A_{0}、B_{0}、C_{0}$ 分别为边 $B C、C A、A B$ 的中点。考虑穿过 $H$ 的三个圆：$A_{0} 周围的$\omega_{a}$、$B_{0}$周围的 \omega_{b}$ 和 $C_{0}$ 周围的 $\omega_{c}$。圆 $\omega_{a}$ 与线 $B C$ 相交于 $A_{1}$ 和 $A_{2} ； \omega_{b}$ 与 $C A$ 相交于 $B_{1}$ 和 $B_{2} ； \omega_{c}$ 与 $A B$ 相交于 $C_{1}$ 和 $C_{2}$。证明点 $A_{1}、A_{2}、B_{1}、B_{2}、C_{1}、C_{2}$ 位于圆上。