The "mature" way to think of this is as a function on the Riemann sphere - this the standard complex plane with a point $\infty$ added, also called the "extended complex plane". The function $1/z$ is a special case of what is called a linear fractional transformation - these are all functions of the form $\frac{az+b}{cz+d}$, where the coefficients are in $\mathbb{C}$. One property of linear fractional transformations is that they send "generalized circles" to "generalized circles", where a generalized circle is a either a normal circle or a line (a line is a called a generalized circle because you can view it as a circle through the point at infinity).
In this case, we see our circle through 0 will be sent through a "circle through infinity" because we define $1/0$ to be $\infty$ here. So your four points will be sent to a line, and in particular, $1/z_1$, $1/z_2$, and $1/z_3$ will be collinear.
Linear fractional transformations and their properties are a standard topic in complex analysis, and you don't even need to know any analysis to prove the property I just mentioned. Any standard text on complex analysis should have the proof. I suggest you find a copy of Gamelin's book, if you are interested.