On a property of plane curves

Abstract

Let $\gamma: [0,1] \to [0,1]^2$ be a continuous curve such that $\gamma(0)=(0,0)$, $\gamma(1)=(1,1)$, and $\gamma(t) \in (0,1)^2$ for all $t\in (0,1)$. We prove that, for each $n \in \mathbb{N}$, there exists a sequence of points $A_i$, $0\leq i \leq n+1$, on $\gamma$ such that $A_0=(0,0)$, $A_{n+1}=(1,1)$, and the sequences $\pi_1(\overrightarrow{A_iA_{i+1}})$ and $\pi_2(\overrightarrow{A_iA_{i+1}})$, $0\leq i \leq n$, are positive and the same up to order, where $\pi_1,\pi_2$ are projections on the axes.