Lawn overrun with weeds
 

My lawn is currently overrun with the following weeds which are taking up more space than the grass itself. Since this spring they are back with a vengeance and overgrowing the grass. They proved completely resistant to heavy treatment with weedkiller last year. Mowing consists mostly of cutting these little blighters down, for them to overgrow any grass after.

I enclose a scan of this nasty specimen. Can anyone identify this weed, and suggest means of getting rid of it? Would I need to dig up my entire lawn and plant seed from scratch?