  The story of positive geometry of massless scalar theories was pioneered in
[1] in the context of bi-adjoint $\phi^3$ theories. Further study proposed that
the positive geometry for a generic massless scalar theory with polynomial
interaction is a class of polytopes called accordiohedra [2]. Tree-level planar
scattering amplitudes of the theory can be obtained from a weighted sum of the
canonical forms of the accordiohedra. In this paper, using results of the
recent work [3], we show that in theories with polynomial interactions all the
weights can be determined from the factorization property of the
accordiohedron. We also extend the projective recursion relations introduced in
[4,5] to these theories. We then give a detailed analysis of how the recursion
relations in $\phi^p$ theories and theories with polynomial interaction
correspond to projective triangulations of accordiohedra. Following the very
recent development [6] we also extend our analysis to one-loop integrands in
the quartic theory.
