I think that I've come to a new approach/theory to prove the congruence of two triangles:

"Triangles are congruent if two pairs of corresponding sides and a non-included angle are equal in both triangles while the distance between the vertexes of the other two pairs of uncorresponding angles are equal."

This is a proof: (*The Question is: Is this a new approach/theorem or it's false?) enter image description here

*The two-headed arrow $YB$, was just added as a construction for the proof, but it's not actually required.

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    $\begingroup$ I'm willing to believe it's a true theorem. Note however that it's different from traditional congruence theorems such as SSS and SAS: those theorems depend only upon measurements of the triangles themselves, while your statement depends in addition upon where the triangles have been placed in the plane. $\endgroup$ – Greg Martin Oct 15 '14 at 1:52

This is a proof that depends on where the triangles are placed in the plane. More specifically, if the condition that two of the edges do not lie in the same line is not held, then this is usually false.

Consider the simple counterexample where C and X coincide whereas A and Z do not. This also means that BCYZ do not lie on the same line. Your condition would not hold, but the triangles may still be congruent.

Hope that helps :)

  • $\begingroup$ It does help, Thanks, Sean. :) And I understand your point, however, I see that this should be considered a bit trivial since it could be fixed by a basic Euclidean transformation/rotation.. $\endgroup$ – Ahmed Oct 15 '14 at 2:54
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    $\begingroup$ This may be trivial when there are only 2 triangles in the figure. In more complicated geometric figures where there are extra conditions on the vertices of the triangle, forcing them to be fixed in some positions relative to other vertices of the triangles, then the transformation would not preserve the restrictions. $\endgroup$ – Sean Lo Oct 15 '14 at 3:07

First of all, Your theorem is correct and so is your proof.

As Sean pointed out, your theorem is very restrictive Since it requires some conditions on where the triangles locate in the plane. You mentioned that we can deal with that by transformations and rotations and so on. Ok, This could work well, but in mathematics, there are some questions to be answered when we face a new theorem.

For example, Are the conditions the theorem requires necessary? are they only sufficient? are they both?

If the later is the case then this is a great news! this doesn't mean that if the conditions are only sufficient but not necessary, that this is bad, not at all. What I mean is that if we have some sufficient and necessary conditions then this is better.

For your theorem, your conditions are only "sufficient" but not necessarly required. We have already some theorems whose conditions are both, necessary and sufficient.

Having said that, the question is, Do we really need this particular theorem? Are there any problems of importance which can't be solved easily without your theorem? If this is the case, I think your efforts will be appreciated, otherwise, why to take your theorem into consideration at all if we have "better" tools to do everything your theorem does?

I mean, Why to consider this as a theorem not as an exercise in any geometry book? You should recognise that "theorems" in mathematics are the important facts which are essential to understand the mathematical releam So not everything we prove is a "theorem".


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