Convex lattice pentagon with three pairs of parallel sides and diagonals

This paper investigates convex lattice pentagons with at least three pairs (a i , d i ), where a i d i , i.e., diagonals parallel to sides.Based on the given conditions, we will form a system of Diophantine equations whose solutions we seek within the set of natural numbers or positive rational numbers.To characterize all obtained convex lattice pentagons of minimal area, we will use the concept of integer unimodular transformations.Specifically, these transformations of the plane preserve the parallelism of lattice segments, the number of lattice points inside a convex lattice polygon and on its boundary, as well as its area.We will then determine the minimum area of the pentagon in each resulting class and identify the pentagon with the smallest diameter.Finally, we will determine all convex lattice pentagons in which three sides are respectively parallel to three diagonals.