MathDB

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_{32}V_{22}}

are circumscribed, then the curved quadrilateral

\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}

is circumscribed, too.
Proposed by Géza Kós, Hungary
[asy] pathpen=black; size(400); pair A=(0,0), B=(4,0), C=(10,0); draw(L(A,C,0.3)); MP("A",A); MP("B",B); MP("C",C); pair X=(5,-7); path G1=D(arc(X,C,A)); pair Y=(5,7), Z=(9,6); draw(Z--B--Y); struct T {pair C;real r;}; T f(pair X, pair B, pair Y, pair Z) { pair S=unit(Y-B)+unit(Z-B); real s=abs(sin(angle((Y-B)/(Z-B))/2)); real t=10, r=abs(X-A); pair Q; for(int k=0;k<30;++k) { Q=B+t*S; t-=(abs(X-Q)-r)/abs(S)-s*t; }
T T=new T; T.C=Q; T.r=s*t*abs(S); return T; } void g(pair Q, real r) { real t=0; for(int k=0;k<30;++k) { X=(5,t); t+=(abs(X-Q)+r-abs(X-A)); } } pair Z1=(1.07,6); draw(B--Z1); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G2=D(arc(X,C,A)); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G3=D(arc(X,C,A)); pen p=black+fontsize(8); MC("\gamma_1",G1,0.85,p); MC("\gamma_2",G2,0.85,NNW,p); MC("\gamma_3",G3,0.85,WNW,p); MC("h_1",B--Z1,0.95,E,p); MC("h_2",B--Y,0.95,E,p); MC("h_3",B--Z,0.95,E,p); path[] G={G1,G2,G3}; path[] H={B--Z1,B--Y,B--Z}; pair[][] al={{S+SSW,S+SSW,3*S},{SE,NE,NW},{2*SSE,2*SSE,2*E}}; for(int i=0;i<3;++i) for(int j=0;j<3;++j) MP("V_{"+string(i+1)+string(j+1)+"}",IP(H,G[j]),al[j],fontsize(8));[/asy]

Problem Statement