Triangles inscribed in smooth closed arcs

by

(Cluj)
Concerning a more general problem stated by p. TURAN (see in [2]), E. G. STRAUS has proved the following theorem (unpublished):

Given any continuum $\mathrm{\Sigma }(=$$\mathrm{\Sigma }(=$Sigma(=\Sigma(=$\mathrm{\Sigma }(=$ compact connected set $)$$)$))$)$ in the Euclidean plane ${\mathbf{R}}^2$${\mathbf{R}}^2$R^(2)\mathbf{R}^{2}${\mathbf{R}}^2$, such that for some triangle $ABC$$ABC$ABCA B C$ABC$ in the plane no three points $A^{\prime },B^{\prime },C^{\prime }$$A^{\prime },B^{\prime },C^{\prime }$A^('),B^('),C^(')A^{\prime}, B^{\prime}, C^{\prime}$A^{\prime },B^{\prime },C^{\prime }$ in $\mathrm{\Sigma }$$\mathrm{\Sigma }$Sigma\Sigma$\mathrm{\Sigma }$ form a triangle similar to $ABC$$ABC$ABCA B C$ABC$, then $\mathrm{\Sigma }$$\mathrm{\Sigma }$Sigma\Sigma$\mathrm{\Sigma }$ is a simple arc.

From this theorem in particular follows that if $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ is a closed arc, then there exists a triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$$A^{\prime }{B}^{\prime }{C}^{\prime }$A^(')B^(')C^(')A^{\prime} B^{\prime} C^{\prime}$A^{\prime }{B}^{\prime }{C}^{\prime }$ similar to $ABC$$ABC$ABCA B C$ABC$, which is inscribed in $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ in the sense that $A^{\prime },B^{\prime },C^{\prime }\in \mathrm{\Gamma }$$A^{\prime },B^{\prime },C^{\prime }\in \mathrm{\Gamma }$A^('),B^('),C^(')in GammaA^{\prime}, B^{\prime}, C^{\prime} \in \Gamma$A^{\prime },B^{\prime },C^{\prime }\in \mathrm{\Gamma }$.

We ask now about the existence of a triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$$A^{\prime }{B}^{\prime }{C}^{\prime }$A^(')B^(')C^(')A^{\prime} B^{\prime} C^{\prime}$A^{\prime }{B}^{\prime }{C}^{\prime }$ inscribed in the closed arc $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$, which has parallel sides with the sides of the triangle $ABC$$ABC$ABCA B C$ABC$ and the same orientation. This special case appears in some problems in the geometry of convex sets (see our notes [1] and [2]). The answer in general may be negative. In our note we solve this problem in the special case when $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ is a closed arc of class $C^1$$C^1$C^(1)C^{1}$C^1$ and has additional conditions about its tangent lines. From our theorem we derive a characterization of the strictly convex closed arcs of the class $C^1$$C^1$C^(1)C^{1}$C^1$.

Le mm a 1. Let be $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega$\mathrm{\Omega }$ the strip formed by the points $(x,y)$$(x,y)$(x,y)(x, y)$(x,y)$ of the Euclidean plane ${\mathbf{R}}^2$${\mathbf{R}}^2$R^(2)\mathbf{R}^{2}${\mathbf{R}}^2$ with $a\leqq x\leqq b(a<b)$$a\leqq x\leqq b(a<b)$a <= x <= b(a < b)a \leqq x \leqq b(a<b)$a\leqq x\leqq b(a<b)$. Denote by ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ and ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$ two simple arcs of the class $C^1$$C^1$C^(1)C^{1}$C^1$ in $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega$\mathrm{\Omega }$, with disjoint interiors, having the parametric representations given by

and suppose that $\phi (0)=\xi (0)=a$$\phi (0)=\xi (0)=a$varphi(0)=xi(0)=a\varphi(0)=\xi(0)=a$\phi (0)=\xi (0)=a$ and $\phi (1)=\xi (1)=b$$\phi (1)=\xi (1)=b$varphi(1)=xi(1)=b\varphi(1)=\xi(1)=b$\phi (1)=\xi (1)=b$. Suppose that ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ and ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$ have a finite number of points in which the tangent is parallel to $Oy$$Oy$OyO y$Oy$
and in the neighbourhood of which the arc is on the same side of the tangent line. Then:
(i) There exists a set of segments $T^{\prime }(t)T^{\prime \prime }(t)$$T^{\prime }(t)T^{\prime \prime }(t)$T^(')(t)T^('')(t)T^{\prime}(t) T^{\prime \prime}(t)$T^{\prime }(t)T^{\prime \prime }(t)$ with the endpoints $T^{\prime }(t)\in {\mathrm{\Gamma }}^{\prime }$$T^{\prime }(t)\in {\mathrm{\Gamma }}^{\prime }$T^(')(t)inGamma^(')T^{\prime}(t) \in \Gamma^{\prime}$T^{\prime }(t)\in {\mathrm{\Gamma }}^{\prime }$ and $T^{\prime \prime }(t)\in {\mathrm{\Gamma }}^{\prime \prime }$$T^{\prime \prime }(t)\in {\mathrm{\Gamma }}^{\prime \prime }$T^('')(t)inGamma^('')T^{\prime \prime}(t) \in \Gamma^{\prime \prime}$T^{\prime \prime }(t)\in {\mathrm{\Gamma }}^{\prime \prime }$, depending continuously on the parameter $t\in [0,1]$$t\in [0,1]$t in[0,1]t \in[0,1]$t\in [0,1]$, such that for any $t$$t$tt$t$ the segment is parallel to $Oy$$Oy$OyO y$Oy$ and $T^{\prime }(0)=(\phi (0),\psi (0)),T^{\prime }(1)==(\phi (1),\psi (1))$$T^{\prime }(0)=(\phi (0),\psi (0)),T^{\prime }(1)==(\phi (1),\psi (1))$T^(')(0)=(varphi(0),psi(0)),T^(')(1)==(varphi(1),psi(1))T^{\prime}(0)=(\varphi(0), \psi(0)), T^{\prime}(1)= =(\varphi(1), \psi(1))$T^{\prime }(0)=(\phi (0),\psi (0)),T^{\prime }(1)==(\phi (1),\psi (1))$ and $T^{\prime \prime }(0)=(\xi (0),\eta (0)),T^{\prime \prime }(1)=(\xi (1),\eta (1))$$T^{\prime \prime }(0)=(\xi (0),\eta (0)),T^{\prime \prime }(1)=(\xi (1),\eta (1))$T^('')(0)=(xi(0),eta(0)),T^('')(1)=(xi(1),eta(1))T^{\prime \prime}(0)=(\xi(0), \eta(0)), T^{\prime \prime}(1)=(\xi(1), \eta(1))$T^{\prime \prime }(0)=(\xi (0),\eta (0)),T^{\prime \prime }(1)=(\xi (1),\eta (1))$.
(ii) If $T^{\prime }\in {\mathrm{\Gamma }}^{\prime }$$T^{\prime }\in {\mathrm{\Gamma }}^{\prime }$T^(')inGamma^(')T^{\prime} \in \Gamma^{\prime}$T^{\prime }\in {\mathrm{\Gamma }}^{\prime }$ is arbitrary, then there exists a point $T^{\prime \prime }\in {\mathrm{\Gamma }}^{\prime \prime }$$T^{\prime \prime }\in {\mathrm{\Gamma }}^{\prime \prime }$T^('')inGamma^('')T^{\prime \prime} \in \Gamma^{\prime \prime}$T^{\prime \prime }\in {\mathrm{\Gamma }}^{\prime \prime }$ such that $T^{\prime }=T^{\prime }(t)$$T^{\prime }=T^{\prime }(t)$T^(')=T^(')(t)T^{\prime}=T^{\prime}(t)$T^{\prime }=T^{\prime }(t)$ and $T^{\prime \prime }=T^{\prime \prime }(t)$$T^{\prime \prime }=T^{\prime \prime }(t)$T^('')=T^('')(t)T^{\prime \prime}=T^{\prime \prime}(t)$T^{\prime \prime }=T^{\prime \prime }(t)$ for some $t$$t$tt$t$ in $[0,1]$$[0,1]$[0,1][0,1]$[0,1]$.
(iii) There exist neighbourhoods $U^{\prime }$$U^{\prime }$U^(')U^{\prime}$U^{\prime }$ and $U^{\prime \prime }$$U^{\prime \prime }$U^('')U^{\prime \prime}$U^{\prime \prime }$ of $T^{\prime }(0)$$T^{\prime }(0)$T^(')(0)T^{\prime}(0)$T^{\prime }(0)$ and $T^{\prime \prime }(0)$$T^{\prime \prime }(0)$T^('')(0)T^{\prime \prime}(0)$T^{\prime \prime }(0)$, such that if $T^{\prime }\in {\mathrm{\Gamma }}^{\prime }\cap U^{\prime }$$T^{\prime }\in {\mathrm{\Gamma }}^{\prime }\cap U^{\prime }$T^(')inGamma^(')nnU^(')T^{\prime} \in \Gamma^{\prime} \cap U^{\prime}$T^{\prime }\in {\mathrm{\Gamma }}^{\prime }\cap U^{\prime }$ and $T^{\prime \prime }\in {\mathrm{\Gamma }}^{\prime \prime }\cap U^{\prime \prime }$$T^{\prime \prime }\in {\mathrm{\Gamma }}^{\prime \prime }\cap U^{\prime \prime }$T^('')inGamma^('')nnU^('')T^{\prime \prime} \in \Gamma^{\prime \prime} \cap U^{\prime \prime}$T^{\prime \prime }\in {\mathrm{\Gamma }}^{\prime \prime }\cap U^{\prime \prime }$ and $T^{\prime }{T}^{\prime \prime }$$T^{\prime }{T}^{\prime \prime }$T^(')T^('')T^{\prime} T^{\prime \prime}$T^{\prime }{T}^{\prime \prime }$ is parallel to $Oy$$Oy$OyO y$Oy$, then there exists a $t$$t$tt$t$ such that $T^{\prime }=T^{\prime }(t)$$T^{\prime }=T^{\prime }(t)$T^(')=T^(')(t)T^{\prime}=T^{\prime}(t)$T^{\prime }=T^{\prime }(t)$ and $T^{\prime \prime }=T^{\prime \prime }(t)$$T^{\prime \prime }=T^{\prime \prime }(t)$T^('')=T^('')(t)T^{\prime \prime}=T^{\prime \prime}(t)$T^{\prime \prime }=T^{\prime \prime }(t)$.

Proof. We proceed by induction on the number $m$$m$mm$m$ of points of ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ and ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$ (different from their endpoints), in which the tangent is parallel to $Oy$$Oy$OyO y$Oy$ and in the neighbourhood of which the ares are on the same side of the tangent lines. (The number of these points is obviously even.)

Suppose that $m=0$$m=0$m=0m=0$m=0$. In this case the arcs ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ and ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$ may be represented in the form

Then the set of segments $T^{\prime }(t)T^{\prime \prime }(t)$$T^{\prime }(t)T^{\prime \prime }(t)$T^(')(t)T^('')(t)T^{\prime}(t) T^{\prime \prime}(t)$T^{\prime }(t)T^{\prime \prime }(t)$ where $t=\frac{x-a}{b-a}$$t=\frac{x-a}{b-a}$t=(x-a)/(b-a)t=\frac{x-a}{b-a}$t=\frac{x-a}{b-a}$ and

satisfies all the conditions of the lemma.
Suppose that the lemma holds for $m\le 2n$$m\le 2n$m <= 2nm \leq 2 n$m\le 2n$ and prove it for $m=2(n+1)$$m=2(n+1)$m=2(n+1)m=2(n+1)$m=2(n+1)$.
Let be $T_k^{\prime }\in {\mathrm{\Gamma }}^{\prime }$$T_k^{\prime }\in {\mathrm{\Gamma }}^{\prime }$T_(k)^(')inGamma^(')T_{k}^{\prime} \in \Gamma^{\prime}$T_k^{\prime }\in {\mathrm{\Gamma }}^{\prime }$, the point in which the tangent is parallel to $Oy$$Oy$OyO y$Oy$ and has the minimal abscissa relative to all the points with this property in the interior of ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ and ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$. Suppose that $T_k^{\prime }=(\phi \left(t_k^{\prime }\right),\psi \left(t_k^{\prime }\right))$$T_k^{\prime }=\left(\phi \left(t_k^{\prime }\right),\psi \left(t_k^{\prime }\right)\right)$T_(k)^(')=(varphi(t_(k)^(')),psi(t_(k)^(')))T_{k}^{\prime}=\left(\varphi\left(t_{k}^{\prime}\right), \psi\left(t_{k}^{\prime}\right)\right)$T_k^{\prime }=(\phi \left(t_k^{\prime }\right),\psi \left(t_k^{\prime }\right))$ and let be $T_i^{\prime }==(\phi \left(t_i^{\prime }\right),\psi \left(t_i^{\prime }\right)),i=1,\dots ,k\to 1$$T_i^{\prime }==\left(\phi \left(t_i^{\prime }\right),\psi \left(t_i^{\prime }\right)\right),i=1,\dots ,k\to 1$T_(i)^(')==(varphi(t_(i)^(')),psi(t_(i)^('))),i=1,dots,k rarr1T_{i}^{\prime}= =\left(\varphi\left(t_{i}^{\prime}\right), \psi\left(t_{i}^{\prime}\right)\right), i=1, \ldots, k \rightarrow 1$T_i^{\prime }==(\phi \left(t_i^{\prime }\right),\psi \left(t_i^{\prime }\right)),i=1,\dots ,k\to 1$ all the points with $t_i^{\prime }<t_k^{\prime }$$t_i^{\prime }<t_k^{\prime }$t_(i)^(') < t_(k)^(')t_{i}^{\prime}<t_{k}^{\prime}$t_i^{\prime }<t_k^{\prime }$, and with the property that the tangents to ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ are parallel to $Oy$$Oy$OyO y$Oy$. Suppose that $T_i^{\prime }==(\phi \left(t_l^{\prime }\right),\psi \left(t_l^{\prime }\right)),1\leqq l\leqq k-1$$T_i^{\prime }==\left(\phi \left(t_l^{\prime }\right),\psi \left(t_l^{\prime }\right)\right),1\leqq l\leqq k-1$T_(i)^(')==(varphi(t_(l)^(')),psi(t_(l)^('))),1 <= l <= k-1T_{i}^{\prime}= =\left(\varphi\left(t_{l}^{\prime}\right), \psi\left(t_{l}^{\prime}\right)\right), 1 \leqq l \leqq k-1$T_i^{\prime }==(\phi \left(t_l^{\prime }\right),\psi \left(t_l^{\prime }\right)),1\leqq l\leqq k-1$ is a point with the maximal abscissa relative to the set of points $T_i^{\prime },i=1,\dots ,k-1$$T_i^{\prime },i=1,\dots ,k-1$T_(i)^('),i=1,dots,k-1T_{i}^{\prime}, i=1, \ldots, k-1$T_i^{\prime },i=1,\dots ,k-1$, (see Fig. 1). For $i\le k$$i\le k$i <= ki \leq k$i\le k$ we shall denote by $t_i^{\prime \prime }$$t_i^{\prime \prime }$t_(i)^('')t_{i}^{\prime \prime}$t_i^{\prime \prime }$ the minimal value of $t^{\prime \prime }$$t^{\prime \prime }$t^('')t^{\prime \prime}$t^{\prime \prime }$ with the property that $\xi \left(t_i^{\prime \prime }\right)=\phi \left(t_i^{\prime }\right)$$\xi \left(t_i^{\prime \prime }\right)=\phi \left(t_i^{\prime }\right)$xi(t_(i)^(''))=varphi(t_(i)^('))\xi\left(t_{i}^{\prime \prime}\right)=\varphi\left(t_{i}^{\prime}\right)$\xi \left(t_i^{\prime \prime }\right)=\phi \left(t_i^{\prime }\right)$. Put $T_i^{\prime \prime }=(\xi \left(t_i^{\prime \prime }\right),\eta \left(t_i^{\prime \prime }\right))$$T_i^{\prime \prime }=\left(\xi \left(t_i^{\prime \prime }\right),\eta \left(t_i^{\prime \prime }\right)\right)$T_(i)^('')=(xi(t_(i)^('')),eta(t_(i)^('')))T_{i}^{\prime \prime}=\left(\xi\left(t_{i}^{\prime \prime}\right), \eta\left(t_{i}^{\prime \prime}\right)\right)$T_i^{\prime \prime }=(\xi \left(t_i^{\prime \prime }\right),\eta \left(t_i^{\prime \prime }\right))$. Then the segments $T_i^{\prime }{T}_i^{\prime \prime },i=1,\dots ,k$$T_i^{\prime }{T}_i^{\prime \prime },i=1,\dots ,k$T_(i)^(')T_(i)^(''),i=1,dots,kT_{i}^{\prime} T_{i}^{\prime \prime}, i=1, \ldots, k$T_i^{\prime }{T}_i^{\prime \prime },i=1,\dots ,k$ are parallel to Oy.

Consider now the strip $a\leqq x\leqq \phi \left(t_l^{\prime }\right)$$a\leqq x\leqq \phi \left(t_l^{\prime }\right)$a <= x <= varphi(t_(l)^('))a \leqq x \leqq \varphi\left(t_{l}^{\prime}\right)$a\leqq x\leqq \phi \left(t_l^{\prime }\right)$, and the part ${\mathrm{\Gamma }}_1^{\prime }$${\mathrm{\Gamma }}_1^{\prime }$Gamma_(1)^(')\Gamma_{1}^{\prime}${\mathrm{\Gamma }}_1^{\prime }$ of ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ for $0\leqq \le t^{\prime }\leqq t_i^{\prime }$$0\leqq \le t^{\prime }\leqq t_i^{\prime }$0≦≤t^(') <= t_(i)^(')0 \leqq \leq t^{\prime} \leqq t_{i}^{\prime}$0\leqq \le t^{\prime }\leqq t_i^{\prime }$ and the part ${\mathrm{\Gamma }}_1^{\prime \prime }$${\mathrm{\Gamma }}_1^{\prime \prime }$Gamma_(1)^('')\Gamma_{1}^{\prime \prime}${\mathrm{\Gamma }}_1^{\prime \prime }$ of ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$ for $0\leqq t^{\prime \prime }\leqq t_l^{\prime \prime }$$0\leqq t^{\prime \prime }\leqq t_l^{\prime \prime }$0 <= t^('') <= t_(l)^('')0 \leqq t^{\prime \prime} \leqq t_{l}^{\prime \prime}$0\leqq t^{\prime \prime }\leqq t_l^{\prime \prime }$. The number of points on ${\mathrm{\Gamma }}_1^{\prime }$${\mathrm{\Gamma }}_1^{\prime }$Gamma_(1)^(')\Gamma_{1}^{\prime}${\mathrm{\Gamma }}_1^{\prime }$ and ${\mathrm{\Gamma }}_1^{\prime \prime }$${\mathrm{\Gamma }}_1^{\prime \prime }$Gamma_(1)^('')\Gamma_{1}^{\prime \prime}${\mathrm{\Gamma }}_1^{\prime \prime }$ in which the tangents are parallel to $Oy$$Oy$OyO y$Oy$ is $\le 2n$$\le 2n$<= 2n\leq 2 n$\le 2n$, because two such points, the point $T_k^{\prime }$$T_k^{\prime }$T_(k)^(')T_{k}^{\prime}$T_k^{\prime }$ and $T_l^{\prime }$$T_l^{\prime }$T_(l)^(')T_{l}^{\prime}$T_l^{\prime }$, are eliminated ( $T_k^{\prime }$$T_k^{\prime }$T_(k)^(')T_{k}^{\prime}$T_k^{\prime }$ is eliminated because $t^{\prime }>t_1^{\prime }$$t^{\prime }>t_1^{\prime }$t^(') > t_(1)^(')t^{\prime}>t_{1}^{\prime}$t^{\prime }>t_1^{\prime }$ and then $T^{\prime }\notin {\mathrm{\Gamma }}_1^{\prime }$$T^{\prime }\notin {\mathrm{\Gamma }}_1^{\prime }$T^(')!inGamma_(1)^(')T^{\prime} \notin \Gamma_{1}^{\prime}$T^{\prime }\notin {\mathrm{\Gamma }}_1^{\prime }$ and $T_1^{\prime }$$T_1^{\prime }$T_(1)^(')T_{1}^{\prime}$T_1^{\prime }$ is eliminated $$$${ }$$ is bes $T_k>T_l$$T_k>T_l$T_(k) > T_(l)T_{k}>T_{l}$T_k>T_l$ and $T_l$$T_l$T_(l)T_{l}$T_l$ is eliminated because it is an endpoint of ${\mathrm{\Gamma }}_1$${\mathrm{\Gamma }}_1$Gamma_(1)\Gamma_{1}${\mathrm{\Gamma }}_1$.) By the induction hypothesis there exists then the set of segments $T^{\prime }(t)T^{\prime \prime }(t)$$T^{\prime }(t)T^{\prime \prime }(t)$T^(')(t)T^('')(t)T^{\prime}(t) T^{\prime \prime}(t)$T^{\prime }(t)T^{\prime \prime }(t)$ with the propeties (i), (ii) and (iii) with respect to the $\mathrm{arcs}{\mathrm{\Gamma }}_1^{\prime }$$\mathrm{arcs}{\mathrm{\Gamma }}_1^{\prime }$arcsGamma_(1)^(')\operatorname{arcs} \Gamma_{1}^{\prime}$\mathrm{arcs}{\mathrm{\Gamma }}_1^{\prime }$ and ${\mathrm{\Gamma }}_1^{\prime \prime }$${\mathrm{\Gamma }}_1^{\prime \prime }$Gamma_(1)^('')\Gamma_{1}^{\prime \prime}${\mathrm{\Gamma }}_1^{\prime \prime }$.

By a similar way may be seen the existence of a set of segments $T^{\prime }(t)T^{\prime \prime }(t)$$T^{\prime }(t)T^{\prime \prime }(t)$T^(')(t)T^('')(t)T^{\prime}(t) T^{\prime \prime}(t)$T^{\prime }(t)T^{\prime \prime }(t)$ with the properties (i), (ii) and (iii) for the parts ${\mathrm{\Gamma }}_2^{\prime }$${\mathrm{\Gamma }}_2^{\prime }$Gamma_(2)^(')\Gamma_{2}^{\prime}${\mathrm{\Gamma }}_2^{\prime }$ of ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ for

$t_l^{\prime }\leqq t^{\prime }\leqq t_k^{\prime }$$t_l^{\prime }\leqq t^{\prime }\leqq t_k^{\prime }$t_(l)^(') <= t^(') <= t_(k)^(')t_{l}^{\prime} \leqq t^{\prime} \leqq t_{k}^{\prime}$t_l^{\prime }\leqq t^{\prime }\leqq t_k^{\prime }$ and ${\mathrm{\Gamma }}_2^{\prime \prime }$${\mathrm{\Gamma }}_2^{\prime \prime }$Gamma_(2)^('')\Gamma_{2}^{\prime \prime}${\mathrm{\Gamma }}_2^{\prime \prime }$ of ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$ for $t_k^{\prime \prime }\leqq t^{\prime \prime }\leqq t_l^{\prime \prime }$$t_k^{\prime \prime }\leqq t^{\prime \prime }\leqq t_l^{\prime \prime }$t_(k)^('') <= t^('') <= t_(l)^('')t_{k}^{\prime \prime} \leqq t^{\prime \prime} \leqq t_{l}^{\prime \prime}$t_k^{\prime \prime }\leqq t^{\prime \prime }\leqq t_l^{\prime \prime }$. (Here we have $t_k^{\prime \prime }<t_l^{\prime }$$t_k^{\prime \prime }<t_l^{\prime }$t_(k)^('') < t_(l)^(')t_{k}^{\prime \prime}<t_{l}^{\prime}$t_k^{\prime \prime }<t_l^{\prime }$, from the definition of the points $T_k^{\prime \prime }$$T_k^{\prime \prime }$T_(k)^('')T_{k}^{\prime \prime}$T_k^{\prime \prime }$ and $T_l^{\prime \prime }$$T_l^{\prime \prime }$T_(l)^('')T_{l}^{\prime \prime}$T_l^{\prime \prime }$, because $\xi \left(t_k^{\prime \prime }\right)<\xi \left(t_l^{\prime \prime }\right)$$\xi \left(t_k^{\prime \prime }\right)<\xi \left(t_l^{\prime \prime }\right)$xi(t_(k)^('')) < xi(t_(l)^(''))\xi\left(t_{k}^{\prime \prime}\right)<\xi\left(t_{l}^{\prime \prime}\right)$\xi \left(t_k^{\prime \prime }\right)<\xi \left(t_l^{\prime \prime }\right)$.)

We may consider now the part ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ of ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ and ${\mathrm{\Gamma }}_3^{\prime \prime }$${\mathrm{\Gamma }}_3^{\prime \prime }$Gamma_(3)^('')\Gamma_{3}^{\prime \prime}${\mathrm{\Gamma }}_3^{\prime \prime }$ of ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$ for $t_k^{\prime }\leqq t^{\prime }\leqq 1$$t_k^{\prime }\leqq t^{\prime }\leqq 1$t_(k)^(') <= t^(') <= 1t_{k}^{\prime} \leqq t^{\prime} \leqq 1$t_k^{\prime }\leqq t^{\prime }\leqq 1$ and $t_k^{\prime \prime }\leqq t^{\prime \prime }\leqq 1$$t_k^{\prime \prime }\leqq t^{\prime \prime }\leqq 1$t_(k)^('') <= t^('') <= 1t_{k}^{\prime \prime} \leqq t^{\prime \prime} \leqq 1$t_k^{\prime \prime }\leqq t^{\prime \prime }\leqq 1$ respectively, and apply the induction hypothesis for ${\mathrm{\Gamma }}_3^{\prime }$${\mathrm{\Gamma }}_3^{\prime }$Gamma_(3)^(')\Gamma_{3}^{\prime}${\mathrm{\Gamma }}_3^{\prime }$ and ${\mathrm{\Gamma }}_3^{\prime \prime }$${\mathrm{\Gamma }}_3^{\prime \prime }$Gamma_(3)^('')\Gamma_{3}^{\prime \prime}${\mathrm{\Gamma }}_3^{\prime \prime }$.

As a final step, by a simple joining of the obtained families of segments $T^{\prime }(t)T^{\prime \prime }(t)$$T^{\prime }(t)T^{\prime \prime }(t)$T^(')(t)T^('')(t)T^{\prime}(t) T^{\prime \prime}(t)$T^{\prime }(t)T^{\prime \prime }(t)$ we obtain a family of segments which completes the proof of the assertion for $m=2(n+1)$$m=2(n+1)$m=2(n+1)m=2(n+1)$m=2(n+1)$. This completes the proof of the lemma.
theorem 1. Let be ABC a triangle in the Euclidean plane ${\mathbf{R}}^2$${\mathbf{R}}^2$R^(2)\mathbf{R}^{2}${\mathbf{R}}^2$. Suppose that $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ is a simple closed arc of class $C^1$$C^1$C^(1)C^{1}$C^1$ in ${\mathbf{R}}^2$${\mathbf{R}}^2$R^(2)\mathbf{R}^{2}${\mathbf{R}}^2$, which has the property that for at least one of the sides of $ABC$$ABC$ABCA B C$ABC$ there is a finite number of points of $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$, in which the tangents are parallel to the respective side and in the neighbourhood of which the arc is on the same side of the tangent line. Then there exists a triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$$A^{\prime }{B}^{\prime }{C}^{\prime }$A^(')B^(')C^(')A^{\prime} B^{\prime} C^{\prime}$A^{\prime }{B}^{\prime }{C}^{\prime }$ with sides parallel to sides of $ABC$$ABC$ABCA B C$ABC$ and of the same orientation as $ABC$$ABC$ABCA B C$ABC$, which is inscribed in $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$, in the sense that $A^{\prime }$$A^{\prime }$A^(')A^{\prime}$A^{\prime }$, $B^{\prime },C^{\prime }\in \mathrm{\Gamma }$$B^{\prime },C^{\prime }\in \mathrm{\Gamma }$B^('),C^(')in GammaB^{\prime}, C^{\prime} \in \Gamma$B^{\prime },C^{\prime }\in \mathrm{\Gamma }$.

Proof. Suppose that $BC$$BC$BCB C$BC$ is the side for which there are a finite number of points in which $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ has tangents parallel to $BC$$BC$BCB C$BC$ and in the neighbourhood of which the arc is on the same side of the tangent line, and let $BC$$BC$BCB C$BC$ be parallel to $Oy$$Oy$OyO y$Oy$. Let be $a$$a$aa$a$ and $b,a<b$$b,a<b$b,a < bb, a<b$b,a<b$ the minimum, respective the maximum of abscissas of the points in which $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ has tangents parallel to $Oy$$Oy$OyO y$Oy$. Denote by ${\mathrm{\Delta }}^{\prime }$${\mathrm{\Delta }}^{\prime }$Delta^(')\Delta^{\prime}${\mathrm{\Delta }}^{\prime }$ and ${\mathrm{\Delta }}^{\prime \prime }$${\mathrm{\Delta }}^{\prime \prime }$Delta^('')\Delta^{\prime \prime}${\mathrm{\Delta }}^{\prime \prime }$ the lines $x=a$$x=a$x=ax=a$x=a$, respectively $x=b$$x=b$x=bx=b$x=b$ and let be $P^{\prime }\in {\mathrm{\Delta }}^{\prime }\cap \mathrm{\Gamma }$$P^{\prime }\in {\mathrm{\Delta }}^{\prime }\cap \mathrm{\Gamma }$P^(')inDelta^(')nn GammaP^{\prime} \in \Delta^{\prime} \cap \Gamma$P^{\prime }\in {\mathrm{\Delta }}^{\prime }\cap \mathrm{\Gamma }$ and $P^{\prime \prime }\in {\mathrm{\Delta }}^{\prime \prime }\cap \mathrm{\Gamma }$$P^{\prime \prime }\in {\mathrm{\Delta }}^{\prime \prime }\cap \mathrm{\Gamma }$P^('')inDelta^('')nn GammaP^{\prime \prime} \in \Delta^{\prime \prime} \cap \Gamma$P^{\prime \prime }\in {\mathrm{\Delta }}^{\prime \prime }\cap \mathrm{\Gamma }$. The points $P^{\prime }$$P^{\prime }$P^(')P^{\prime}$P^{\prime }$ and $P^{\prime \prime }$$P^{\prime \prime }$P^('')P^{\prime \prime}$P^{\prime \prime }$ divide the are $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ in two ares: ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ and ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$.

Consider now the triangle $ABC$$ABC$ABCA B C$ABC$ and let be $D$$D$DD$D$ the foot of the perpendicular from $A$$A$AA$A$ to the supporting line of $BC$$BC$BCB C$BC$. Then $D=qB+(1-q)C$$D=qB+(1-q)C$D=qB+(1-q)CD=q B+(1-q) C$D=qB+(1-q)C$ for a given number $q$$q$qq$q$. Denote by $r$$r$rr$r$ the vector $A-D$$A-D$A-DA-D$A-D$. Then $A=D+r$$A=D+r$A=D+rA=D+r$A=D+r$.

Let be $T^{\prime }{T}^{\prime \prime }$$T^{\prime }{T}^{\prime \prime }$T^(')T^('')T^{\prime} T^{\prime \prime}$T^{\prime }{T}^{\prime \prime }$ a segment parallel to $BC$$BC$BCB C$BC$. The triangle $T{T}^{\prime }{T}^{\prime \prime }$$T{T}^{\prime }{T}^{\prime \prime }$TT^(')T^('')T T^{\prime} T^{\prime \prime}$T{T}^{\prime }{T}^{\prime \prime }$ having parallel sides with the sides of $ABC$$ABC$ABCA B C$ABC$ and the same orientation, may be determined by putting

We proceed now to construction of an arc $\gamma$$\gamma$gamma\gamma$\gamma$ in the following way:
According to Lemma 1, the segment $T^{\prime }(0)T^{\prime \prime }(0)(=P^{\prime })$$T^{\prime }(0)T^{\prime \prime }(0)\left(=P^{\prime }\right)$T^(')(0)T^('')(0)(=P^('))T^{\prime}(0) T^{\prime \prime}(0)\left(=P^{\prime}\right)$T^{\prime }(0)T^{\prime \prime }(0)(=P^{\prime })$ can be moved continuously to $T^{\prime }(1)T^{\prime \prime }(1)(=P^{\prime \prime })$$T^{\prime }(1)T^{\prime \prime }(1)\left(=P^{\prime \prime }\right)$T^(')(1)T^('')(1)(=P^(''))T^{\prime}(1) T^{\prime \prime}(1)\left(=P^{\prime \prime}\right)$T^{\prime }(1)T^{\prime \prime }(1)(=P^{\prime \prime })$ such that all the intermediate positions are segments $T^{\prime }{T}^{\prime \prime }$$T^{\prime }{T}^{\prime \prime }$T^(')T^('')T^{\prime} T^{\prime \prime}$T^{\prime }{T}^{\prime \prime }$ parallel to $Oy$$Oy$OyO y$Oy$, and $T^{\prime }\in {\mathrm{\Gamma }}^{\prime },T^{\prime \prime }\in {\mathrm{\Gamma }}^{\prime \prime }$$T^{\prime }\in {\mathrm{\Gamma }}^{\prime },T^{\prime \prime }\in {\mathrm{\Gamma }}^{\prime \prime }$T^(')inGamma^('),T^('')inGamma^('')T^{\prime} \in \Gamma^{\prime}, T^{\prime \prime} \in \Gamma^{\prime \prime}$T^{\prime }\in {\mathrm{\Gamma }}^{\prime },T^{\prime \prime }\in {\mathrm{\Gamma }}^{\prime \prime }$. Consider this family of segments $T^{\prime }{T}^{\prime \prime }$$T^{\prime }{T}^{\prime \prime }$T^(')T^('')T^{\prime} T^{\prime \prime}$T^{\prime }{T}^{\prime \prime }$ depending continuously on the parameter $t\in \in [0,1]$$t\in \in [0,1]$t∈∈[0,1]t \in \in[0,1]$t\in \in [0,1]$, i.e. suppose that the endpoints $T^{\prime }$$T^{\prime }$T^(')T^{\prime}$T^{\prime }$ and $T^{\prime \prime }$$T^{\prime \prime }$T^('')T^{\prime \prime}$T^{\prime \prime }$ depends continuously on $t$$t$tt$t$. Then by (1), $T$$T$TT$T$ depends continuously on $t$$t$tt$t$. Denote

We have obviously $\gamma (0)=T(0)=P^{\prime }$$\gamma (0)=T(0)=P^{\prime }$gamma(0)=T(0)=P^(')\gamma(0)=T(0)=P^{\prime}$\gamma (0)=T(0)=P^{\prime }$ and $\gamma (1)=T(1)=P^{\prime \prime }$$\gamma (1)=T(1)=P^{\prime \prime }$gamma(1)=T(1)=P^('')\gamma(1)=T(1)=P^{\prime \prime}$\gamma (1)=T(1)=P^{\prime \prime }$.
According to the theorem of Jordan, $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ determines two domains of ${\mathbf{R}}^2$${\mathbf{R}}^2$R^(2)\mathbf{R}^{2}${\mathbf{R}}^2$, which we will call the "inside" and the „outside" of $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$. In what follows we shall prove that there exists a value $t_0$$t_0$t_(0)t_{0}$t_0$ in $(0,1)$$(0,1)$(0,1)(0,1)$(0,1)$ such that $T\left(t_0\right)$$T\left(t_0\right)$T(t_(0))T\left(t_{0}\right)$T\left(t_0\right)$ is inside and a value $t_1$$t_1$t_(1)t_{1}$t_1$ in $(0,1)$$(0,1)$(0,1)(0,1)$(0,1)$ such that $T\left(t_1\right)$$T\left(t_1\right)$T(t_(1))T\left(t_{1}\right)$T\left(t_1\right)$ is outside $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$.

We observe that from the condition of the theorem it follows that $P^{\prime }$$P^{\prime }$P^(')P^{\prime}$P^{\prime }$ is an isolated point with the property that the tangent to $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ is parallel to $Oy$$Oy$OyO y$Oy$. Let be $T^{\prime }$$T^{\prime }$T^(')T^{\prime}$T^{\prime }$ a point on ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ and let be $T^{\prime \prime }$$T^{\prime \prime }$T^('')T^{\prime \prime}$T^{\prime \prime }$ the point on ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$ with minimal value of the parameter $t^{\prime \prime }$$t^{\prime \prime }$t^('')t^{\prime \prime}$t^{\prime \prime }$ such that $T^{\prime }{T}^{\prime \prime }$$T^{\prime }{T}^{\prime \prime }$T^(')T^('')T^{\prime} T^{\prime \prime}$T^{\prime }{T}^{\prime \prime }$ is parallel to $Oy$$Oy$OyO y$Oy$. Let be $T^{\prime \prime }{R}^{\prime }$$T^{\prime \prime }{R}^{\prime }$T^('')R^(')T^{\prime \prime} R^{\prime}$T^{\prime \prime }{R}^{\prime }$ and $T^{\prime }{R}^{\prime \prime }$$T^{\prime }{R}^{\prime \prime }$T^(')R^('')T^{\prime} R^{\prime \prime}$T^{\prime }{R}^{\prime \prime }$ the segments parallel to $CA$$CA$CAC A$CA$ and $BA$$BA$BAB A$BA$ respectively. Letting $T^{\prime }\to P^{\prime }$$T^{\prime }\to P^{\prime }$T^(')rarrP^(')T^{\prime} \rightarrow P^{\prime}$T^{\prime }\to P^{\prime }$, the segment $T^{\prime }{T}^{\prime \prime }$$T^{\prime }{T}^{\prime \prime }$T^(')T^('')T^{\prime} T^{\prime \prime}$T^{\prime }{T}^{\prime \prime }$ tends to $P^{\prime }$$P^{\prime }$P^(')P^{\prime}$P^{\prime }$. From the property that $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ is of class $C^1$$C^1$C^(1)C^{1}$C^1$, it follows that the parallel line in $P^{\prime }$$P^{\prime }$P^(')P^{\prime}$P^{\prime }$ to $BA$$BA$BAB A$BA$ will intersect $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ in a point. Denote by $Q^{\prime \prime }$$Q^{\prime \prime }$Q^('')Q^{\prime \prime}$Q^{\prime \prime }$ the nearest point to $P^{\prime }$$P^{\prime }$P^(')P^{\prime}$P^{\prime }$ with this property. Then the open segment $P^{\prime }{Q}^{\prime \prime }$$P^{\prime }{Q}^{\prime \prime }$P^(')Q^('')P^{\prime} Q^{\prime \prime}$P^{\prime }{Q}^{\prime \prime }$ will be inside $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$. Similarly, let $P^{\prime }{Q}^{\prime }$$P^{\prime }{Q}^{\prime }$P^(')Q^(')P^{\prime} Q^{\prime}$P^{\prime }{Q}^{\prime }$ be the segment with the same property, parallel to $CA$$CA$CAC A$CA$. Let be $T_1^{\prime }$$T_1^{\prime }$T_(1)^(')T_{1}^{\prime}$T_1^{\prime }$ a point on ${\mathrm{\Gamma }}^{\prime }$${\mathrm{\Gamma }}^{\prime }$Gamma^(')\Gamma^{\prime}${\mathrm{\Gamma }}^{\prime }$ and denote by $T_1^{\prime \prime }$$T_1^{\prime \prime }$T_(1)^('')T_{1}^{\prime \prime}$T_1^{\prime \prime }$ the point on ${\mathrm{\Gamma }}^{\prime \prime }$${\mathrm{\Gamma }}^{\prime \prime }$Gamma^('')\Gamma^{\prime \prime}${\mathrm{\Gamma }}^{\prime \prime }$ with minimal value of the parameter $t^{\prime \prime }$$t^{\prime \prime }$t^('')t^{\prime \prime}$t^{\prime \prime }$, such that $T_1^{\prime }{T}_1^{\prime \prime }$$T_1^{\prime }{T}_1^{\prime \prime }$T_(1)^(')T_(1)^('')T_{1}^{\prime} T_{1}^{\prime \prime}$T_1^{\prime }{T}_1^{\prime \prime }$ is parallel te $Oy$$Oy$OyO y$Oy$. Suppose that the abscissa of $T_1^{\prime }\leqq$$T_1^{\prime }\leqq$T_(1)^(') <=T_{1}^{\prime} \leqq$T_1^{\prime }\leqq$ minimum of the abscissas of $Q^{\prime }$$Q^{\prime }$Q^(')Q^{\prime}$Q^{\prime }$ and $Q^{\prime \prime }$$Q^{\prime \prime }$Q^('')Q^{\prime \prime}$Q^{\prime \prime }$. Then $T_1^{\prime }{T}_1^{\prime \prime }$$T_1^{\prime }{T}_1^{\prime \prime }$T_(1)^(')T_(1)^('')T_{1}^{\prime} T_{1}^{\prime \prime}$T_1^{\prime }{T}_1^{\prime \prime }$ will intersect $P^{\prime }{Q}^{\prime }$$P^{\prime }{Q}^{\prime }$P^(')Q^(')P^{\prime} Q^{\prime}$P^{\prime }{Q}^{\prime }$ in $R^{\prime }$$R^{\prime }$R^(')R^{\prime}$R^{\prime }$ and $P^{\prime }{Q}^{\prime \prime }$$P^{\prime }{Q}^{\prime \prime }$P^(')Q^('')P^{\prime} Q^{\prime \prime}$P^{\prime }{Q}^{\prime \prime }$ in $R^{\prime \prime }$$R^{\prime \prime }$R^('')R^{\prime \prime}$R^{\prime \prime }$. Because $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ is a simple closed arc, for $T_1^{\prime }$$T_1^{\prime }$T_(1)^(')T_{1}^{\prime}$T_1^{\prime }$ sufficiently near