  The absolute bolometric luminosity of the point of core Helium ignition in
old, metal poor, red-giant stars is of roughly constant magnitude, varying only
very slightly with mass or metallicity. It can thus be used as a standard
candle. Here, we review the main difficulties in measuring this location in any
real dataset and develop an empirical approach to optimise it for tip of the
red giant branch (TRGB) analysis. We go on to present a new algorithm for the
identification of the TRGB in nearby metal poor stellar systems. Our method
uses a least-squares fit of a data-adaptive slope to the luminosity function in
1 magnitude windows. This finds the region of the luminosity function that
shows the most significant decline in star counts as we go to brighter
magnitudes; the base of this decline is attributed as the location of the tip.
This technique then allows for the determination of realistic uncertainties
which reflect the quality of the luminosity function used, but which are
typically ~0.02 mags rms + ~ 0.03 mags systematic, a significant improvement
upon previous methods that have used the tip as a standard candle. Finally, we
apply our technique to the Local Group spiral galaxy M33 and the dwarf galaxies
And I & II, and derive distance modulii of 24.50 +/- 0.06 (794 +/- 23 kpc),
24.33 +/- 0.07 (735 +/- 23 kpc) and 24.05 +/- 0.06 (645 +/- 19 kpc)
respectively. The result for M33 is in excellent agreement with the Cepheid
distances to this galaxy, and makes the possibility of a significant amount of
reddening in this object unlikely.
