Geometric Criteria and Focal Reconstruction for Cassini Point Sets: An Elementary Euclidean Approach

This preprint gives necessary-and-sufficient criteria for finite point sets in the real Euclidean plane to admit two foci with a common positive product of distances, together with reconstruction of the admissible focal pairs. For six distinct points, the criterion consists of three simultaneous polynomial conditions of degree at most four, an explicit degeneracy test, and checks at at most three exceptional midpoints. Each admissible midpoint determines exactly one unordered focal pair. The argument proceeds from elementary midpoint and equal-product relations, through circle-or-line classifications, to a square-image transformation that converts focal reconstruction into a genuine-circle problem. It treats collinear and cyclic quadruples, rectangles, and orthocentric systems, while retaining the real-existence conditions, converse branches, and exceptional cases needed for a complete criterion. The contribution is methodological as well as mathematical. Pythagoras' theorem, similarity, midpoint relations, and circle powers are not merely preliminary tools awaiting replacement by advanced machinery: appropriately organized, they can carry the essential argument. Complex numbers provide compact notation for planar relations, and determinants encode elementary elimination and concyclicity conditions; their appearance does not entail a reliance on complex analysis or advanced algebraic theory. The graph formulation requires only the propagation of equality along paths. No differentiation or analysis is used. Escalating the theoretical machinery before examining the elementary structure is a methodological risk. Neither the degree of an equation nor the sophistication of the surrounding literature determines the tools a particular problem requires. Neglecting elementary relations can impose an unnecessary technical barrier and leave an accessible route to a solution unexplored. The scope is finite Cassini point sets and the circle-target condition discussed in polynomial inscription; no extension to arbitrary Jordan target curves is claimed. The upload contains the revised English manuscript as a reading PDF and an editable Word document with MathType equations. The revision adds a methodological discussion while leaving the 25 theorems, proofs, formulae, and references unchanged.