Finding centerline of peninsula?

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Given a peninsula stretching north south and the two coast lines connecting in the south, I want to find out which is the place, or rather line, which is furthest away from both coastlines on the peninsula? What I am looking for is the red line in the below image. The black and pink lines are the eastern and western coastlines.



One way could be to generate a polygon and do a negative buffer. An other one could be to add the longitudinal values, and divide by two. The latter one seems more rational - still it demands some programming and coordinate handling.

But perhaps there is a third way? Any suggestions? ArcGIS and QGIS are both relevant tools of the trade here.



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