The experimental determination of the quantum critical point (QCP) that triggers the self-organization of charged striped domains in cuprate perovskites is reported. The phase diagram of doped cuprate superconductors is determined by a first variable, the hole doping δ, and a second variable, the micro-strain ε of the Cu-O bond length, obtained from the Cu K-edge extended x-ray absorption fine structure. For a fixed optimum doping, δ c = 0.16, we show the presence of the QCP for the onset of local lattice distortions and stripe formation at the critical micro-strain ε c. The critical temperature T c (ε, δ) reaches its maximum at the quantum critical point (ε c , δ c) for the formation of bubbles of superconducting stripes. The critical charge, orbital and spin fluctuations near this strain QCP provide the interaction for the pairing.