Some Research On4-isosceles Planar Sets

A finite planar set P is k-isosceles for k≥3if every k-point subset of the set P contains a point equidistant from two others.In1998, P. Fishburn discussed4-isosceles sets, and provided some partial results about4-isosceles sets.This thesis discusses further4-isosceles sets. By the definition of the4-isosceles sets, every n-1subset of the4-isosceles set with n points for n>4is also4-isosceles. The basis of the research is to characterize all the4-isosceles4-point sets, and therefore we start with4-isosceles4-point sets. We obtain the following results:(1) We classify all the4-isosceles4-point sets, according to the number of isosceles trian-gles included. Based on the classification, we provide the necessary and sufficient conditions of how to determine the4-isosceles4-point sets.(2) We discuss the4-isosceles5-point sets which exactly contain nine or eight isosceles triangles, which is based on the classification obtained above.(3) We characterize a convex4-isosceles6-point set with certain isomorphism class.(4) We obtain three new4-isosceles9-point sets.