Three circular arcs
![\gamma_1, \gamma_2,](/media/m/9/d/e/9deafad5e76a70f90b315885383c4219.png)
and
![\gamma_3](/media/m/5/1/c/51c4016a2310b853d2c6304f512b686a.png)
connect the points
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![C.](/media/m/4/6/7/467f2e8003bd034885e63601825c1836.png)
These arcs lie in the same half-plane defined by line
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
in such a way that arc
![\gamma_2](/media/m/3/e/a/3ea4ab248844cc703cc3f2f700b8c0ec.png)
lies between the arcs
![\gamma_1](/media/m/4/2/0/420f82fcb29ad57c285227afe9841fd5.png)
and
![\gamma_3.](/media/m/2/a/1/2a1fbc325b25e2aae137585569409b35.png)
Point
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
lies on the segment
![AC.](/media/m/c/d/7/cd70da53daefabfd42b8ee7939775882.png)
Let
![h_1, h_2](/media/m/e/3/a/e3ac3a6e370932ef349b74b35c20b9e3.png)
, and
![h_3](/media/m/2/d/3/2d322048c3839657a6b6f35329bfb154.png)
be three rays starting at
![B,](/media/m/1/6/e/16e519ccc501d3fbc4fe4ab09a16195c.png)
lying in the same half-plane,
![h_2](/media/m/5/d/d/5dd34799df33b7962251de451023a5a5.png)
being between
![h_1](/media/m/4/0/5/405fe34684b6152379543cc1eecd9f00.png)
and
![h_3.](/media/m/0/c/3/0c38364007c00c4701a055fdabee5f9c.png)
For
![i, j = 1, 2, 3,](/media/m/2/f/d/2fda6c7c3297b3f3c487bf560fa3c549.png)
denote by
![V_{ij}](/media/m/b/1/6/b16fd01bef8a5b22e1091d5eefc792f1.png)
the point of intersection of
![h_i](/media/m/b/d/c/bdc138a50ffab17e59a7f0635b9fa02d.png)
and
![\gamma_j](/media/m/c/4/5/c458ec4e7618e51c6eb2d1ca606ea6d9.png)
(see the Figure below). Denote by
![\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}](/media/m/8/3/5/83505956c8f9c17ed8ad356b5fcd584c.png)
the curved quadrilateral, whose sides are the segments
![V_{kj}V_{kl}](/media/m/9/4/e/94eae676275a5eb8a4d328de67b4d458.png)
and arcs
![V_{ij}V_{kj}](/media/m/7/f/9/7f9ff9713a3203f3dd473c01935c5cba.png)
and
![V_{il}V_{kl}.](/media/m/5/a/d/5adc51b18dabe2467c1cd9adb76a41b2.png)
We say that this quadrilateral is
![circumscribed](/media/m/1/c/9/1c92177f2a137c0af9a00a231da220e3.png)
if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals {{ INVALID LATEX }} are circumscribed, then the curved quadrilateral
![\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}](/media/m/a/5/d/a5d56d088abd458b30148b603db16a74.png)
is circumscribed, too.
Proposed by Géza Kós, Hungary
%V0
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 $AC$ in such a way that arc $\gamma_2$ lies between the arcs $\gamma_1$ and $\gamma_3.$ Point $B$ lies on the segment $AC.$ 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 $V_{ij}$ the point of intersection of $h_i$ and $\gamma_j$ (see the Figure below). Denote by $\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}$ the curved quadrilateral, whose sides are the segments $V_{ij}V_{il},$ $V_{kj}V_{kl}$ and arcs $V_{ij}V_{kj}$ and $V_{il}V_{kl}.$ 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 $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{...$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$ is circumscribed, too.
Proposed by Géza Kós, Hungary