2010 IMO Shortlist G7

Three circular arcs $\gamma_{1}, \gamma_{2}$, and $\gamma_{3}$ connect the points $A$ and $C$. These arcs lie in the same half-plane defined by line $A C$ in such a way that arc $\gamma_{2}$ lies between the arcs $\gamma_{1}$ and $\gamma_{3}$. Point $B$ lies on the segment $A C$. Let $h_{1}, h_{2}$, and $h_{3}$ be three rays starting at $B$, lying in the same half-plane, $h_{2}$ being between $h_{1}$ and $h_{3}$. For $i, j=1,2,3$, denote by $V_{i j}$ the point of intersection of $h_{i}$ and $\gamma_{j}$ (see the Figure below). Denote by $\widehat{V_{i j} V_{k j}} \widehat{V_{k \ell} V_{i \ell}}$ the curved quadrilateral, whose sides are the segments $V_{i j} V_{i \ell}, V_{k j} V_{k \ell}$ and $\operatorname{arcs} V_{i j} V_{k j}$ and $V_{i \ell} V_{k \ell}$. We say that this quadrilateral is circumscribed if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals $\sqrt{V_{11} V_{21}} \sqrt{22} V_{12}, \sqrt{V_{12} V_{22}} \sqrt{23} V_{13}, \sqrt{V_{21} V_{31}} \sqrt{V_{32} V_{22}}$ are circumscribed, then the curved quadrilateral $\sqrt{V_{22} V_{32}} \sqrt{33} V_{23}$ is circumscribed, too. Fig. 1 (Hungary)

三个圆弧 $\gamma_{1}、\gamma_{2}$ 和 $\gamma_{3}$ 连接点 $A$ 和 $C$。这些弧位于由线 $A C$ 定义的同一半平面中，使得弧 $\gamma_{2}$ 位于弧 $\gamma_{1}$ 和 $\gamma_{3}$ 之间。点 $B$ 位于线段 $A C$ 上。令 $h_{1}、h_{2}$ 和 $h_{3}$ 为从 $B$ 开始、位于同一半平面内的三条射线，$h_{2}$ 位于 $h_{1}$ 和 $h_{3}$ 之间。对于$i, j=1,2,3$，用$V_{i j}$表示$h_{i}$和$\gamma_{j}$的交点(见下图)。 $\widehat{V_{i j} V_{k j}} \widehat{V_{k \ell} V_{i \ell}}$ 表示弯曲四边形，其边是线段 $V_{i j} V_{i \ell}、V_{k j} V_{k \ell}$ 和 $\operatorname{arcs} V_{i j} V_{k j}$ 和 $V_{i \ell} V_{k \ell}$。如果存在一个圆接触这两个线段和两条弧，我们就说这个四边形是外接四边形。证明如果弯曲四边形 $\sqrt{V_{11} V_{21}} \sqrt{22} V_{12}, \sqrt{V_{12} V_{22}} \sqrt{23} V_{13}, \sqrt{V_{21} V_{31}} \sqrt{V_{32} V_{22}}$ 为外接，则弯曲四边形 $\sqrt{V_{22} V_{32}} \sqrt{33} V_{23}$ 也是外接的。图1(匈牙利)