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This paper introduces advances in the geometry of the ratio of either two orthree points in a line in the Desargues affine plane, and we see this as aratio of elements of skew field which are constructed over a line in Desarguesaffine plane. The results given here have a clean, geometric presentation basedDesargues affine plan axiomatics and definitions of addition and multiplicationof points on a line in this plane, and for skew field properties. The resultsin this paper are: (1) study of properties for ratio of two and three points,in a line on Desargues affine plane. Also, we discuss the cases related to the"line-skew field" characteristic, when it is two and when it is different fromtwo. (2) we have construct the maps for ratio points-set, for two and threepoints, and have prove that, this maps are bijections of the lines. (3) set ofratio points (for two and for three points) with addition and multiplication ofpoints, forms a skew fields, for more, this skew fields are sub-skew fields ofthe 'line-skew field' on Desargues affine plane. (4) Every Dyck polygoncontaining co-linear ratio vertices in the Desargues affine plane has a freegroup presentation.

Advances in the Geometry of the Ratio of Linear Points in the Desargues Affine Plane Skew Field