Ceci n'est pas un engrenage? :)

Because the gear movement vectors add up to a vector equal to the applied force, but opposite in direction :-D Remember, the magnitude and direction of a vector are the sum of the magnitudes and directions of the vector components. If you applied a given force to any of the gears, in any direction, you could tabulate the resulting forces and directions and add them up. What you will have then, for the image, is a four quantities of equal magnitude but where the odd numbered quantities have opposite sign than the even number quantities. Classic parallelogram.

Say, you applied a force to the green gear to move it clockwise, let's call it +N (for positive force of magnitude N; we chose +ve to mean clockwise.) All four gears are of equal size, so they have identical torques; they will all move by the same speed. Taking friction as negligible, we can expect any gear to produce a force of the same magnitude as that applied to it, but with opposite direction.

So back to the green gear, when we apply a clock wise force of magnitude N, it gives us back a counter clockwise for of N, which acts of the yellow gear (or the red one, if we choose to.) Yellow one is acted upon by +N, and it in turn gives us -N back (a counter clockwise one.) Feed that to the blue one and you get +N, which acts on red to give us -N.

Now, a force of -N is exerted by red upon green; BUT! we are applying a force of +N to green! What's going on?

The two forces cancel out and you have nothing.