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Yeah well, in high school we're taught that we can prove two triangles to be congruent using one of those five criteria: SSS, SAS, ASA, AAS and HL.

But I'm wondering: Since if a triangle is congruent, everything is congruent (including the length of heights, medians, etc.), are there more ways to prove two triangles are congruent?

For example, suppose I know two triangles are similar, and the length of the angle bisector of the largest angle of those triangles is the same. Is this sufficient to say that they both are congruent?

Deathkamp Drone
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    Yes. Any similarity transformations mapping one of the triangles onto the other, maps the angle bisector from the largest angle to the angle bisector of the other largest angle. If those have the same length, the similarity factor equals $1$ and the triangles are indeed congruent. – user133281 Mar 09 '14 at 22:40
  • Okay, thank you. I don't know what should I do with this question now, delete it or leave it that way? – Deathkamp Drone Mar 10 '14 at 17:17
  • @CarpeNoctem This question is way too vast. Have you considered how many possibilities are you creating, with those parameters. I tried answering, but got overwhelmed. For example, can you prove that three heights give us a congruence criteria? Now think of cases like two heights and a median. I suggest you deal with individual cases whenever you require them. – Sawarnik Mar 10 '14 at 19:31
  • Yeah, there are many possibilities. I was just wondering if there were any general theorems covering this. I assume that if you're given certain parameters, you can prove that it narrows down to one of the 4 fundamental cases. But as I said, I was just curious if there were more theorems covering congruence between triangles other than those four axioms (?). – Deathkamp Drone Mar 11 '14 at 00:37

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