We perform a 3D multi-probe analysis of the rich galaxy cluster A1689 by combining improved weak-lensing data from new BVRiz Subaru/Suprime-Cam observations with strong-lensing, X-ray, and Sunyaev-Zeldovich effect (SZE) data sets. We reconstruct the projected matter distribution from a joint weak-lensing analysis of 2D shear and azimuthally integrated magnification constraints, the combination of which allows us to break the mass-sheet degeneracy. The resulting mass distribution reveals elongation with axis ratio ~0.7 in projection. When assuming a spherical halo, our full weak-lensing analysis yields a projected concentration of $c_{200c}^{2D}=8.9pm 1.1$ ($c_{vir}^{2D}sim 11$), consistent with and improved from earlier weak-lensing work. We find excellent consistency between weak and strong lensing in the region of overlap. In a parametric triaxial framework, we constrain the intrinsic structure and geometry of the matter and gas distributions, by combining weak/strong lensing and X-ray/SZE data with minimal geometric assumptions. We show that the data favor a triaxial geometry with minor-major axis ratio 0.39+/-0.15 and major axis closely aligned with the line of sight (22+/-10 deg). We obtain $M_{200c}=(1.2pm 0.2)times 10^{15} M_{odot}/h$ and $c_{200c}=8.4pm 1.3$, which overlaps with the $>1sigma$ tail of the predicted distribution. The shape of the gas is rounder than the underlying matter but quite elongated with minor-major axis ratio 0.60+/-0.14. The gas mass fraction within 0.9Mpc is 10^{+3}_{-2}%. The thermal gas pressure contributes to ~60% of the equilibrium pressure, indicating a significant level of non-thermal pressure support. When compared to Plancks hydrostatic mass estimate, our lensing measurements yield a spherical mass ratio of $M_{Planck}/M_{GL}=0.70pm 0.15$ and $0.58pm 0.10$ with and without corrections for lensing projection effects, respectively.
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