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.