  [Abridged] We present the first determination of the intrinsic
three-dimensional shapes and the physical parameters of both dark matter (DM)
and intra-cluster medium (ICM) in a triaxial galaxy cluster. While most
previous studies rely on the standard spherical modeling, our approach allows
to infer the properties of the non-spherical intra-cluster gas distribution
sitting in hydrostatic equilibrium within triaxial DM halos by combining X-ray,
weak and strong lensing observations. We present an application of our method
to the galaxy cluster MACS J1423.8+2404. This source is an example of a well
relaxed object with a unimodal mass distribution and we infer shape and
physical properties of the ICM and the DM for this source. We found that this
is a triaxial galaxy cluster with DM halo axial ratios 1.53+/-0.15 and
1.44+/-0.07 on the plane of the sky and along the line of sight, respectively.
We show that accounting for the three-dimensional geometry allows to solve the
long-standing discrepancy between galaxy cluster masses determined from X-ray
and gravitational lensing observations. We focus also on the determination of
the inner slope of the DM density profile alpha, since the cuspiness of
dark-matter density profiles is one of the critical tests of the cold dark
matter (CDM) paradigm for structure formation: we measure alpha=0.94+/-0.09 by
accounting explicitly for the 3D structure for this cluster, a value which is
close to the CDM predictions, while the standard spherical modeling leads to
the biased value alpha=1.24+/-0.07. Our findings provide further evidences that
support the CDM scenario and open a new window in recovering the intrinsic
shapes and physical parameters of galaxy clusters in a bias-free way. This has
important consequences in using galaxy clusters as cosmological probes.
