Question 22
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 $\widehat{V_{11}V_{21}}\,\widehat{V_{22}V_{12}},\quad\widehat{V_{12}V_{22}}\,\widehat{V_{23}V_{13}},\quad\widehat{V_{21}V_{31}}\,\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\,\widehat{V_{33}V_{23}}$ is circumscribed, too.
Fig. 1
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 $\widehat{V_{11}V_{21}}\,\widehat{V_{22}V_{12}},\quad\widehat{V_{12}V_{22}}\,\widehat{V_{23}V_{13}},\quad\widehat{V_{21}V_{31}}\,\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\,\widehat{V_{33}V_{23}}$ is circumscribed, too.
Fig. 1