  We calculate all planar contributions to the two-loop massless helicity
amplitudes for the process $q\bar q\to \gamma\gamma\gamma$. The results are
presented in fully analytic form in terms of the functional basis proposed
recently by Chicherin and Sotnikov. With this publication we provide the
two-loop contributions already used by us in the NNLO QCD calculation of the
LHC process $pp\to \gamma\gamma\gamma$ [Chawdhry et al. (2019)]. Our results
agree with a recent calculation of the same amplitude [Abreu et al. (2020)]
which was performed using different techniques. We combine several modern
computational techniques, notably, analytic solutions for the IBP identities,
finite-field reconstruction techniques as well as the recent approach [Chen
(2019)] for efficiently projecting helicity amplitudes. Our framework appears
well-suited for the calculation of two-loop multileg amplitudes for which
complete sets of master integrals exist.
