We revisit the inverse problem associated with Electrical Impedance Tomography. This important imaging technique has numerous clinical applications, in particular in pulmonary, cardiac and brain imaging, as well as in the detection of breast cancer. We simplify the derivations of the basic analytical inversion formula. Also, we introduce a novel computational inversion algorithm, which by harvesting powerful analytical and geometrical techniques, yields a drastically faster algorithm than the one arising from the current d-bar approach. The approximate time to reconstruct a typical phantom is reduced from 15 min to 5 s.