The bipolar, nonthermal, high-latitude lobes known as the Fermi bubbles (FBs) are thought to originate from a massive energy release near the Galactic centre (GC). We constrain the FB engine and the circumgalactic medium (CGM) by analytically and numerically modeling the FB edges as strong forward shocks, as inferred from recent observations. A non-directed energy release produces shocks too spherical to account for observations even for a maximally massive Galactic disc, critical CGM rotation, or injection effectively offset from the GC. In contrast, collimated injection nearly perpendicular to the disc can account for observations in both ballistic (free expansion) and slowdown regimes, as we show using a simple stratified evolution model verified by hydrodynamic simulations. FBs still in their ballistic regime require injection (at $zsimeq100$ pc heights in our model) with a half-opening angle $thetasimeq4^circ$, a normalized velocity $beta_{-2}equiv v/(0.01c)gtrsim 0.4$, and an energy $Egtrsim2beta_{-2}^2times 10^{55}$ erg, launched $mathbb{T}simeq 3.3beta_{-2}^{-1}$ Myr ago, showing a distinctive low-pressure region behind the bubble head. Slowing-down (mass accumulated) FBs require a faster injection, a thinner jet, a smaller $E/(beta_{-2}theta)^{2}$, and a comparable $mathbb{T}$, and follow a ballistic stage that must reach a height $z_{s}gtrsim 5$ kpc.
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