OK I'll bite, I'm curious and skeptical of your proof Tom. In order to assume the SSS angle theorem, your theory must assume that the planes fly in straight lines to their destination.  Lets look at this.  I want to see for myself if your argument that the earth is flat holds water.  I took your first 2 itineraries and plotted them approximately on the FE world map posted on this site: Brisbane -> Sydney ->Perth -> Sydney and  Sydney -> London -> Dubai -> Sydney.  So these do indeed create triangles as can be expected from 3 points on a plane.  Looking good so far.  Please feel free to correct my airport placements if they are in error.

Based on your previous response it seems that you agree with 3DGeek that WorldAtlas and Qantas publish approximately accurate travel distances and times. Lets take these distances (same distances you used in your SSS angle calculations) and see what they look like on the map.

For the AUS loop: 2849km, 2649km, 3605km respectively. (Red Dots and lines)  Seems plausible based on my rough approximation of city locations on the FE world map.

Next for the INTL loop: 17174km, 5493km, 12039km respectively. (Yellow Dots and lines) Sydney to London (flight 1) and London to Dubai (flight 2) seem to match up relatively well.  Flight 2 appears to be just less than 1/3 the length of flight 1 if we look at the lengths of the yellow lines.  Flight 3 however doesnt match up, it should be just more than 2/3 the length of flight 1, but it's nearly the same length.  OK, that's a little weird but its difficult to approximate line lengths.  Let's not rely on this for this proof.

To summarize so far, plotting airport locations on an FE map when the airports are in close proximity makes sense but it is difficult to approximate exact line lengths and airport locations on a small map.

Let's continue to dig a little deeper.  Tom let's look at your Triangle Theory. Geometry, that's something I can get behind.  I still want to refer to the map that I attached because I feel it is close enough to prove or disprove the SSS Theorem that Tom offers.

AUS loop:
A = 46.6692 °
B = 81.857 °
C = 51.4738 °
So we have 3 acute angles based on this calculation. Let's look at the map. Looks like one obtuse angle is in there.  Hmm, that's strange.  Anyway let's move to the International loop.  Looking at the map I would have to be way off on my city approximations in order for the angles Tom calculated to be correct.

INTL Loop:
A = 7.77844 °
B = 17.2553 °
C = 154.966 °
Two acute and one VERY obtuse angle. I would have to have at least two of those airports on the wrong CONTINENT if these angles were to match up.

So Tom it seems to me this is very shaky "Proof" that the earth is flat.  The response I'm expecting is that the distances are not correct, that airlines do not fly in straight lines, or that airlines do not fly at cruising speeds.  All of these arguments contradict the assumption your worked on to create the proof you presented earlier in this thread. Unless you can provide a reason why you must contradict yourself I cannot accept it.  3DGeek showed us two independent sources proving flight distances as well as calculations based on facts showing that they are plausible. 

Also, I started this response before you responded to Curious Squirrel, but I'd like to field a response to that one as well; the sum of the angles of a triangle will always be 180 degrees, no more, no less. This is in every math text book ever printed, I won't go through the proof on that I'll leave that burden to you.  The reason your calculations do not add up to 180 every time due to rounding error.  +/-.0003 is not significant enough to base this argument off of.