  In this work, which is the first of a series to prepare a cosmological
parameter analysis with third-order cosmic shear statistics, we model both the
shear three-point correlation functions $\Gamma^{(i)}$ and the third-order
aperture statistics $\langle\mathcal{M}_\mathrm{ap}^3\rangle$ from the
BiHalofit bispectrum model and validate these statistics with a series of
N-body simulations.
  We then investigate how to bin the shear three-point correlation functions to
achieve an unbiased estimate for third-order aperture statistics in real data.
  Finally, we perform a cosmological parameter analysis on KiDS1000-like mock
data with second- and third-order statistics. We recover all cosmological
parameters with very little bias. Furthermore, we find that a joint analysis
almost doubles the constraining power on $S_8$ and increases the
figure-of-merit in the $\Omega_\mathrm{m}$-$\sigma_8$ plane by a factor of 5.9
with respect to an analysis with only second-order shear statistics.
  Our modelling pipeline is publicly available at
https://github.com/sheydenreich/threepoint/releases/.

Cosmology and Nongalactic Astrophysics