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Originally Posted by Breeze
Well, not exactly. As simply as I can describe it, a 3000 lb. car is being pulled toward the earth's surface by gravity. The only thing stopping it from reaching its objective is the equal and opposite force imparted by the four tires. The pressurized air in those tires is pushing the car away from the earth's surface. Given a fixed vehicle mass of 3000 lbs. and a fixed tire inflation pressure of 30 psi, you end up with a fixed total area of tire contact patch needed to equalize the forces. Does that part make sense?
If you add weight to the car but keep the tire pressure the same, the size of the contact patch must neccessarily increase. If you keep the weight of the car the same but deflate the tires, the size of the contact patch must increase in order to keep the forces equal. Hopefully this makes sense as well.
Thus, the area of the contact patch remains contant so long as inflation pressure and the car's weight remain constant. This is true irrespective of the width of the tire. It then just becomes a question of the contact patch's shape, and it is this which determines the car's ultimate handling characteristics.
To be sure, other factors come into play and must be taken into account in determining which tire is best for a given task, but hopefully now the issue of contact patch size is more comprehensible. 
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I'll challenge your description a little. Take a set of 4 inch wide tires from the fifties (18 inchers were on MG Tseries and race cars of that era) and a set of these new 245/45-18 and on two otherwise identical cars start them around a circular track (NASCAR comes to mind) gradually increase speed until each car loses traction and slides into the wall. You are saying both tires would have the same contact patch - I'm saying the car with the larger patch (wider tire)will stick to the pavement better. Another way to think about it is two people that weigh the same and are the same height. One wears size 9, the other size 13's. One has more traction than the other. What about a woman in heels as opposed to flats. Same weight, different psi applied to the ground.