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I wanna know how to do this in QGIS:

enter image description here

I've tried many ways, but still can't find the right solution.

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Consider using Voronoi diagrams. Animated examples appear at mathematica.stackexchange.com/questions/3963/…. – whuber♦ Apr 23 '12 at 15:17

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Don't remember if there is a specific tool, anyway if you need a powerful way to remove overlapping areas then just import your vector in a GRASS mapset, you can do it in the QGIS work environment using the QGIS/GRASS plugin. The GRASS vector model is fully topological so it just does not allow overlapping areas, that during the import are saved in a separate layer. After that you can export the GRASS vector with the "save as..." QGIS dialog.

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Tried answer one -- imported to a GRASS set and then exported back to shapefile with Save As in QGIS -- but it made no difference, the overlapping layers persisted. Is there a step I'm missing? – Tim Henderson Jan 8 at 23:58
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I'm not sure what you need, but from link you shared it looks as full manually operation. In QGIS the same manipulation can be done with Split Features tool.

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