newtonian mechanics - Rigid body dynamics derivation from Newton's laws for higher dimensions

Since Newton's laws are defined for point particles, I'd like to derive some laws of motions for rigid bodies only by considering a rigid body as a system of particles such that the distances from every particle to every other particle doesn't change with time. I think I derived that the force applied on one particle of a rigid body must be the same for every other particle of the rigid body in one dimension by the following:

Consider two particles on a line $P_1$ and $P_2$ both with masses $dm$ and positions $x_1$ and $x_2$. Let's say that a force $F_1$ acts on the particle $P_1$. By Newton's second law we get: $$F_1 = dm\frac{d^2x_1}{dt^2}$$ By the definition of a rigid body, the distance between $P_1$ and $P_2$ doesn't change with time. Define $r$ as this distance ie. $r = x_1 - x_2$. Therefore: $$\frac{dr}{dt} = 0$$ Taking the derivative of both sides we further get that $$\frac{d^2r}{dt^2} = 0$$ $$\frac{d^2(x_1 - x_2)}{dt^2} = 0$$ $$\frac{d^2x_1}{dt^2} = \frac{d^2x_2}{dt^2}$$ By Newton's second law this is the same as: $$\frac{F_1}{dm} = \frac{F_2}{dm}$$ (where $F_2$ is the force acting on $P_2$), and since $dm \ne 0$ finally: $$F_1 = F_2$$

These steps can be done for arbitrary amount of particles, and so we get that in one dimension, if a force is applied on one of the particle of a rigid body, every other particle of the rigid body experiences the same force.

The problem is that I cannot do a similar proof for two dimensions by defining the distance $r = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$, but I'm sure that it can be done, and that if done torque, moment of inertia and center of mass would arise. Can someone do a similar proof for two dimensions, if it can be done like this at all?