Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha, \beta$ linearly independent over $\bQ$. We attach to $\Theta$ a quadruple $\Omega(\Theta)$ of exponents that measure the quality of approximation to $\Theta$ both by rational points and by rational lines. The two ``uniform'' components of $\Omega(\Theta)$ are related by an equation due to Jarn\'\i k, and the four exponents satisfy two inequalities that refine Khintchine's transference principle. Conversely, we show that for any quadruple $\Omega$ fulfilling these necessary conditions, there exists a point $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.

We give a two dimensional extension of the three distance Theorem. Let $\theta$ be in $\mathbf{R}^{2}$ and let $q$ be in $\mathbf{N}$. There exists a triangulation of $\mathbf{R}^{2}$ invariant by $\mathbf{Z}^{2}$-translations, whose set of vertices is $\mathbf{Z}^{2}+\{0,\theta,\dots,q\theta\}$, and whose number of different triangles, up to translations, is bounded above by a constant which does not depend on $\theta$ and $q$.

For each real number $\xi$, let $\lambdahat_2(\xi)$ denote the supremum of all real numbers $\lambda$ such that, for each sufficiently large $X$, the inequalities $|x_0| \le X$, $|x_0\xi-x_1| \le X^{-\lambda}$ and $|x_0\xi^2-x_2| \le X^{-\lambda}$ admit a solution in integers $x_0$, $x_1$ and $x_2$ not all zero, and let $\omegahat_2(\xi)$ denote the supremum of all real numbers $\omega$ such that, for each sufficiently large $X$, the dual inequalities $|x_0+x_1\xi+x_2\xi^2| \le X^{-\omega}$, $|x_1| \le X$ and $|x_2| \le X$ admit a solution in integers $x_0$, $x_1$ and $x_2$ not all zero. Answering a question of Y.~Bugeaud and M.~Laurent, we show that the exponents $\lambdahat_2(\xi)$ where $\xi$ ranges through all real numbers with $[\bQ(\xi)\wcol\bQ]>2$ form a dense subset of the interval $[1/2, (\sqrt{5}-1)/2]$ while, for the same values of $\xi$, the dual exponents $\omegahat_2(\xi)$ form a dense subset of $[2, (\sqrt{5}+3)/2]$. Part of the proof rests on a result of V.~Jarn\'{\i}k showing that $\lambdahat_2(\xi) = 1-\omegahat_2(\xi)^{-1}$ for any real number $\xi$ with $[\bQ(\xi)\wcol\bQ]>2$.

Continued fractions associated with $\GL_3 (\mathbf{Z})$ are introduced and applied to find fundamental units in a two-parameter family of complex cubic fields.