Problem
Let $C\subset\mathbb{R}^2$ be a simple closed convex curve with $O$ as its interior point and its regular parametrization is given as $\gamma_C:[0,1]\rightarrow C$ such that $\gamma_C(0)=(1,0)$ and $\gamma_C'(0)=(0,1)$.
Let $L\subset\mathbb{R}^2$ be a simple curve with its regular parametrization given as $\gamma_L:[0,1]\rightarrow L$ such that $\gamma_L(0)=(1,0)$ and $\gamma_L'(0)=(-1,1)$.
Show that there exists $t\in[0,1]$ such that $\gamma_L(t)$ is interior to $C$.
Description
I am trying to prove a lemma in my research regarding topology but I come across this small problem about geometry of a curve. I do not attend a full course on geometry of curves, hence I may lack of ideas to solve this. It is geometrically intuitive but I need an algebraic approach to prove this. I try Taylor's Theorem or Cauchy's Mean Value Theorem but I'm going nowhere.
Please advice me on what I should do. Thank you in advance.