I've found that kitchen shears are really helpful in this situation.
Place the bird on a cutting board, breast down. Insert the shears into either the neck hole or the other hole. Use your fingers to find the backbone, and cut through the ribs/pelvis just to the side of the backbone, all the way to the other hole. Repeat this to completely remove the backbone, saving the backbone for stock. Lay the bird on the breast, skin-side down. Using both hands, press open the bird so that it lays flat. Using a heavy knife or cleaver, cut through the keel bone in the breast. You may have to use the shears (for the life of me, I can't remember if you have to cut the pelvis a second time, as well. It's been awhile since I did this on my own). You should now have half a chicken, breast, wing, thigh, and leg. Using a Chef's knife, cut the breast/wing section away from the thigh/leg section. You chicken is now quartered.
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