Third-order statistics of the cosmic density field provides a powerful cosmological probe containing synergistic information to the more commonly explored second-order statistics. Here, we exploit a spectroscopic catalog of 72,563 clusters of galaxies extracted from the Sloan Digital Sky Survey, providing the first detection of the baryon acoustic oscillations (BAO) peak in the three-point correlation function (3PCF) of galaxy clusters. We measure and analyze both the connected and the reduced 3PCF of SDSS clusters from intermediate ($rsim10$ Mpc/h) up to large ($rsim140$ Mpc/h) scales, exploring a variety of different configurations. From the analysis of reduced 3PCF at intermediate scales, in combination with the analysis of the two-point correlation function, we constrain both the cluster linear and non-linear bias parameters, $b_1=2.75pm0.03$ and $b_2=1.2pm0.5$. We analyze the measurements of the 3PCF at larger scales, comparing them with theoretical models. The data show clear evidence of the BAO peak in different configurations, which appears more visible in the reduced 3PCF rather than in the connected one. From the comparison between theoretical models considering or not the BAO peak, we obtain a quantitative estimate of this evidence, with a $Delta chi^2$ between 2 and 75, depending on the considered configuration. Finally, we set up a generic framework to estimate the expected signal-to-noise ratio of the BAO peak in the 3PCF exploring different possible definitions, that can be used to forecast the most favorable configurations to be explored also in different future surveys, and applied it to the case of the Euclid mission.
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