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You really know quite a lot.  If it's 0600GMT (approx their morning twilight) and it's your evening twilight, you must be approx 12 hours different, or the other side of the world.  Plot the Ground Points of the three stars as on the attached.  For latitudes up to the give 30 deg N, the distortions aren't outrageous.  Draw the Zenith Distance Circles of Position.  Spica is about the right length since it's North-South.  Since Vega-Pollux intersect somewhere not too far from north of Spica and are mostly East-West, scale the lenghts on this plot by 1/cos(Lat).  So AP could be somewhere near 25N, 157W.

 

But no AP is needed it you're willing to do some math.  The attached spread sheet shows two analytic methods (which, luckily give the same answers).  It can be seen, the 3 intersection points are:  23-18.4N 157-17.5W, 23-18.4N 157-17.5W, and 23-18.3N 157-17.6W