  This paper investigates semilinear elliptic problems in planar triangles with
Dirichlet conditions specified on one side of the boundary and Neumann
conditions imposed on the remaining two sides. By employing moving plane
method, we establish that the positive solution is monotone in the normal
direction of the Dirichlet side when the Neumann vertex is non-obtuse. In the
case where the Neumann vertex is obtuse, the positive solution is monotone in
the normal direction of the longer Neumann side provided some technical
conditions. Furthermore, this monotonicity property extends to the first mixed
eigenfunction in triangles through continuity method via domain deformation. It
is noteworthy that the maximum of the first positive eigenfunction in a
triangle with mixed boundary conditions, consisting of two Neumann sides and
one Dirichlet side, is uniquely located on the Neumann side with the greater
length. This maximum point coincides with the Neumann vertex if and only if
either the Neumann vertex is non-obtuse or the two Neumann sides have equal
lengths. This result successfully resolves a specific problem posed within the
Polymath project: Polymath7 research thread 1.
