The position analysis of planar linkages has been dominated by resultant elimination and tangent-half-angle substitution techniques applied to sets of kinematic loop equations. This analysis is thus reduced to finding the roots of a polynomial in one variable, the characteristic polynomial of the linkage. In this paper, by using a new distance-based technique, it is shown that this standard approach becomes unnecessarily involved when applied to the position analysis of the three seven-link Assur kinematic chains. Indeed, it is shown that the characteristic polynomials of these linkages can be straightforwardly derived without relying on variable eliminations nor trigonometric substitutions, and using no others tools than elementary algebra.