Temporal order of events seen by different Lorentz observers

[spaghetti3451]

Homework Statement

Three events, $A$, $B$, $C$, are seen by observer $\mathcal{O}$ to occur in the order $ABC$. Another observer, $\mathcal{\bar{O}}$, sees the events to occur in the order $CBA$. Is it possible that a third observer sees the events in the order $ACB$? Support your conclusion by drawing a spacetime diagram.

Homework Equations

The Attempt at a Solution

Here's a particular placement of the events $A$, $B$, $C$ in the order $ABC$ as seen by observer $\mathcal{O}$ ($t$-$x$ coordinate system) and in the order $CBA$ as seen by observer $\mathcal{\bar{O}}$ ($t'$-$x'$ coordinate system):

I find that not all placements of the events $A$, $B$, $C$ on the spacetime diagram simultaneously produce the orders $ABC$ and $CBA$ as seen by two different observers $\mathcal{O}$ and $\mathcal{\bar{O}}$.

How I proceed with demonstrating whether or not a third observer sees the events in the order $ACB$?

 

[TSny]

I don't believe you are projecting the events onto the t'-axis correctly.
For example, see https://en.wikipedia.org/wiki/Minkowski_diagram or http://www.phas.ubc.ca/~mav/p200/stnotes.pdf

You are trying to pick a specific choice of events A, B, C on the diagram and a specific choice of observers such that the events have the time ordering asked for in the problem.