A (combinatorial) configuration (

${v}_{3}$) is a partial linear space, with

$v$ points, three points on each line, and three lines passing through each point. For simplicity we identify lines with the set of points incident with them. A (

${v}_{3}$) is called triangle-free if in any triple of noncollinear points there is a pair of points not joinable by a line. This obviously implies its dual. A geometric realization of a (

${v}_{3}$) is an injective collineation

$\varphi $ of (

${v}_{3}$) into the Euclidean plane, i.e.,

$\varphi $ is a map such that points are collinear if and only if their images are. The image of

$\varphi $ is called a geometric (

${v}_{3}$). For triangle-free (

${v}_{3}$) with

$v\le 18$ we have up to isomorphism exactly one (

${15}_{3}$), one (

${17}_{3}$), and four (

${18}_{3}$). The authors show that they all posses geometric realizations. Indeed, the maps are given explicitly. Similarly for the unique point transitive (

${20}_{3}$) and (

${21}_{3}$). The results are computer-generated.