The perimeter-analogue of the centroid of a triangle, when this is viewed as the intersection of the area-bisecting cevians, is the intersection of the perimeter-bisecting cevians, or the Nagel point. The length of the shortest part of the perimeter cut by a variable line through an interior point P of a triangle ranges over a closed interval [w, s], where s is the semiperimeter and \(w=w(P)\) is a real number between 0 and s. When P is the Nagel point, we compute the expression of w(P) in terms of the side lengths and show that the range of w(P)/(2s) over all triangles is the interval (0, 4/9], where 4/9 is attained by the equilateral triangle, and 0 is approached by triangles with sides in bi-ratio approaching 0 : 1 : 1. This result is a perimeter-analogue of the classical Winternitz theorem in a triangle, asserting that, in the case of area-ratio instead of perimeter-ratio and P is the centroid, in every triangle, \(w(P)/\sigma =4/9\) , where \(\sigma \) is the area of the triangle. Hence, the range of \(w(P)/\sigma \) over all triangles is the singleton set \(\{4/9\}\) .
