We present the 3-point function \xi _ { 3 } and Q _ { 3 } = \xi _ { 3 } / \xi _ { 2 } ^ { 2 } for a spectroscopic sample of luminous red galaxies ( LRG ) from the Sloan Digital Sky Survey DR6 & DR7 . We find a strong ( S/N > 6 ) detection of Q _ { 3 } on scales of 55-125 Mpc/h , with a well defined peak around 105 Mpc/h in all \xi _ { 2 } , \xi _ { 3 } and Q _ { 3 } , in excellent agreement with the predicted shape and location of the imprint of the baryon acoustic oscillations ( BAO ) . We use very large simulations ( from a cubic box of L=7680 Mpc/h ) to asses and test the significance of our measurement . Models without the BAO peak are ruled out by the Q _ { 3 } data with 99 % confidence . This detection demonstrates the non-linear growth of structure by gravitational instability between z = 1000 and the present . Our measurements show the expected shape for Q _ { 3 } as a function of the triangular configuration . This provides a first direct measurement of the non-linear mode coupling coefficients of density and velocity fluctuations which , on these large scales , are independent of cosmic time , the amplitude of fluctuations or cosmological parameters . The location of the BAO peak in the data indicates \Omega _ { m } = 0.28 \pm 0.05 and \Omega _ { B } = 0.079 \pm 0.025 ( for h = 0.70 ) after marginalization over spectral index ( n _ { s } = 0.8 - 1.2 ) linear b _ { 1 } and quadratic c _ { 2 } bias , which are found to be in the range : b _ { 1 } = 1.7 - 2.2 and c _ { 2 } = 0.75 - 3.55 . The data allows a hierarchical contribution from primordial non-Gaussianities in the range Q _ { 3 } = 0.55 - 3.35 . These constraints are independent and complementary to the ones that can be obtained using the 2-point function , which are presented in a separate paper . This is the first detection of the shape of Q _ { 3 } on BAO scales , but our errors are shot-noise dominated and the SDSS volume is still relatively small , so there is ample room for future improvement in this type of measurements .