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I have two vector layers, an administrative boundary layer and a roads layer. They look like the following:

Two layers, roads and the administrative boundary

I would like to find the point (within the administrative boundary) that is furthest from a road. I have a spatial analyst licence and I have tried running the Euclidean distance tool but it merely buffers around the coastline.

Do I need to combine both layers first?

Do I need to rasterise them individually and then run the Euclidean distance tool?

I have looked at dozens of other questions on here and I can't seem to find an answer to my problem.

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you need to run the euclidian distance tool with your road layer as input. First check that his layer has a valid coordinate system, prefeably a "local" projected coordinate system. specify a cell size according to your precision (note that if you ask too much, the processing could fail, start with big pixels when you test).

Once you have a good distance layer, you can move to zonal statistics (maximum), there are several post on the topic, e.g. Zonal maximum at a specific location . In short, you will use the raster calculator to find the locations where the distance is equal to the maximum distance

Con("euclidian_dist" == "zonal_max", 1)

Then you convert it to point (raster to point)

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