Planar embeddings of chainable continua

We prove that for a chainable continuum $X$ and every non-zigzag $x\in X$ there exists a planar embedding $\phi:X\to \phi(X)\subset\mathbb R^2$ such that $\phi(x)$ is accessible, partially answering the question of Nadler and Quinn from 1972. Two embeddings $\phi,\psi:X \to \mathbb R^2$ are called strongly equivalent if $\phi \circ \psi^{-1}: \psi(X) \to \phi(X)$ can be extended to a homeomorphism of $\mathbb R^2$. We also prove that every indecomposable chainable continuum can be embedded in the plane in uncountably many strongly non-equivalent ways.