The reasoning for this question comes out of pure curiosity. say if there were, for example 4 bugatti veyron's ( represented by the rectangles on the diagram) all starting from the middle and all attached by rope, and they accelerate out at 90 degrees to each other, reaching a top speed of say 200mph (89.4m/s) before the rope becomes taught (diagram in top right corner) what kind of force will the centre point experience and also the force the driver. assumptions can be made regarding the distance and time, also assuming there are no material limits and ideal situations meaning in the end the centre point does not move and rope joints are infinitely strong. id just like to see how the newtons laws could be applied to a question like this.
The force on each car is purely longitudinal because the restraining force of necessity runs thru the central point (origin). For example, the East and West forces cancel so far as the origin is concerned, so no lateral force exists from the viewpoint of the North or South car.
The restraining force on each car, then, is the same as if it were tied to an unmoveable tiedown point-- the force thus is the torque currently being applied via each car's engine and drivetrain.
The question assumes no material limits, but specifies the very fine Bugatti Veyron, which is made of specific materials. Let's just take two cars driving apart at high speed, joined by an infinitely strong rope. Assuming 89.4m/s and a mass per car of 1,888kg, the total energy of 2*1/2 mv^2 = 15MJ.
If the cars were to come to a halt, this is the amount of energy which would have to be dissipated. This energy could be go into the elongation of the ropes, or more likely into the deformation of the cars. Depending upon what the rope was attached to, either the cars would be ripped apart until the requisite amount of energy has been absorbed by the structure of the car, or the ropes will detach themselves from the car due to local failure of the connection point.
This question is more about energy than balancing forces.
With the information and assumptions you have given, I would say that the forces act in a hyperbolic fashion toward each other, creating a saddle region at the center point as in your figure. To have a rough look see the image below, in which the red arrows indicates the forces acting in a hyperbolic fashion around the center location creating the above-mentioned saddle region.
Hope this helps!!
Edit 1: As J.Ari mentioned, only the net force at (0,0) will be 0N, all the others will behave hyperbolically.