  We report experimental results of the pattern developed by a mixture of two
types of grains in a triangular rotating tumbler operating in the avalanche
regime. At the centroid of the triangular tumbler an invariant zone appears
where the grains do not move relative to the tumbler. We characterize this
invariant zone by its normalized area, $A_i$, and its circularity index as a
function of the normalized filling area $A$. We find a critical filling area so
that only for $A>A_c$ invariant zones are obtained. These zones scale as
$A_i\sim (A-A_c)^2$ near $A_c$. We have obtained a maximum in the circularity
index for $A\approx 0.8$, for which the shape of the invariant zone is closer
to a circular one. The experimental results are reproduced by a simple model
which, based on the surface position, accounts for all the possible straight
lines within the triangle that satisfy the condition of constant $A$. We have
obtained an analytic expression for the contour of the invariant zone.
