You can use a ruler to find the intersection on a grid map (or
maybe just grid paper), or even use some fancy geospacial software
to give it to you. If you're game, you can use a bit of standard
high school maths to work out the formulas that derive the
intersection accurately.
The following geocaches define the two lines. Cache coordinates
are those as given on their web page.
Line 1
Line 2
Because this is a problem in Euclidean geometry (ie, spaces that
are flat, not curved), the geographical Latitude/Longitude
coordinates need to be converted to grid coordinates (where
distances are the same in any direction). New Zealand Transverse
Mercator (NZTM) is now the standard grid projection for New
Zealand. You can
convert to NZTM here, or your GPS unit may even display/convert
NZTM for you.
The
Topo50-BA31 topographic map (shown above) from LINZ is ideal,
as it covers the area and has the NZTM coordinate system
superimposed making it very easy to plot points. You can also use a
large sheet street map.
Convert back to WGS 84 if you can only enter geographical
coordinates into your GPS.
The cache is a 400ml sistema container painted black.
You can check your answers for this puzzle on
Geochecker.com.