  This paper deals with the second Neumann eigenfunction ${u}$ of any planar
triangle ${T}$. In a recent work by C. Judge and S. Mondal [Ann. Math., 2022],
it was established that ${u}$ does not have any critical point within the
interior of ${T}$. In this paper, we show the uniqueness of non-vertex critical
point and the monotonicity property of the second eigenfunction. To be more
precise, when ${T}$ is not an equilateral triangle, the non-vertex critical
point exists if and only if ${T}$ is an acute triangle that is not a
super-equilateral triangle, and the global extrema of ${u}$ are achieved at and
only at the endpoints of the longest side. This establishes the origin theorem
and conjecture 13.6 initially posed by C. Judge and S. Mondal [Ann. Math.,
2020]. Our proof relies heavily on continuity methods, eigenvalue inequalities,
and the maximum principle to establish these results.
