Sharpening Vahlen's result in Diophantine approximation

n this paper we refine Vahlen's 1895 result in Diophantine approximation by providing sharper bounds for the approximation coefficients, especially when at least one of the partial quotients $a_n$ or $a_{n+1}$ of the regular continued fraction expansion $[a_0;a_1,a_2,\dots]$ of $x$ is 1. An improvement of Vahlen's result was already given in papers by Jaroslav Han\u{c}l ([9]), Han\u{c}l and Silvie Bahnerova ([10]), and by Dinesh Sharma Bhattarai ([5]), but the approach of the present paper is very different from Han\u{c}l c.s. We believe that the geometrical methods used in this paper not only offer a significant improvement over Vahlen's result, but also yield new insights that can contribute to improving Borel's classical constant.