Peak location estimation for a triangle wave in AGWN?
You may be able to determine this in the frequency domain. The noise signal is broad band and has low spectral density at any one frequency. The triangle, however, has all energy concentrated in just a few frequencies, so these should stick out in a short term Fourier spectrum. You can match against your expected spectrum (odd harmonics, amplitude proportional to 1/n^2). 

The harmonics of a triangle are all in phase. Since your triangle wave is arbitrarily located inside your analysis window, the harmonics will appear to have a linear phase. You can probably model your measured phase (after proper unwrapping) as a linear phase using a simple least squares error approach. This should give you the exact position of the peak of the triangular wave with respect to the beginning for your analysis window.