I think the problem of existence is not really an issue in when dealing with the Kantian distinction. It is of course important, but it is more of a case of looking at the structure of the propositions in order to determine their truth value. For Kant a priori judgements consist of propositions whose truth can not be determined by reference to any experience but solely on the basis of the terms used in the sentence. For example, 'All green apples are green'. In other to determine the truth value we need only to look at the sentence. Other types of statements that Kant thought were a priori were,"All triangles have three sides" and " 5+2=7". The important point when it comes to a priori judgements is that when we try and negate them this invariably will lead to a contradiction. To claim that all green apples are not actually green is a contradiction. The same argument apples to triangles. To try and deny that a triangle has three sides makes no sense.
A posteriori judgements on the other hand are based on experience and don't involve contradictions. For example, "All living things reproduce", while being true, it is logically consistent to deny the truth of the statement. In other words, if I were to say, "All living things don't reproduce", this wouldn't create a problem in terms of logic. Unlike a priori statements, this makes perfect sense. This doesn't mean it is correct, it just means that it is not contradictory to say so.
Basically we can say that Kant was not happy with this classification. So he came up with the synthetic a priori propositions. He though that judgements could be proven true via their meanings through certain truths about the world. For example, "All bodies are extended". Kant believed that this statement, as the name suggests was both apriori and a posteriori at the same time. Basically we can say he just calls it synthetic apriori. The important point is that is based on experience. The other important point is that we cannot deny the truth of the statement. To do so would lead to some sort of contradiction.
I have a hard time with generalizations, i.e. starting points that are black or white or axioms. It just seems to me that this is what is stopping us from seeing 'truth'. Axioms to me are like starting from an agreed to beginning and then only going forward from there when it is obvious that one can not only go forward but backwards, diagonally, or jump out of the line all together and go any which way.
Now take for example your statement about triangles. Yes, there is a definition to the word triangle that we all have agreed upon. But I don't think we can say that saying a triangle has more than three sides is a contradiction if there is no way to prove there is even such a thing as a perfect triangle in the first place.
I remember watching this program on fractals...http://en.wikipedia.org/wiki/Fractal
...and they were showing how intricate shapes become when broken down into larger images and/or repeated. And I thought to myself, this idea we have of 'truth' may be based in our inability to step outside of aesthetics in order to imagine a different viewpoint than we feel comfortable with.
For example, if we use fractals to look at the point(s) where the lines in a triangle meet that we have magnified, we may see where the angles meet are not an exact angle, but rather a very small line or a series lines which make an acute bend but not actually an angle. If this were the case then we would be fallible to say there was such a thing as a triangle in the first place! This is even more apparent when we consider there may be no such thing as true 'flat' as that anything that has substance or that can be seen and touched (such as a dimensional triangle...as opposed to the abstract thought of a triangle) would at least have some depth and 3D quality to it.
So literally speaking, there could not be a 'true' triangle. There could only be an image our senses want us to 'believe' a true triangle should be. Similar to the way we want to believe in Santa Claus, God, etc. the only difference being is that believing in a triangle is more socially acceptable. Personally, I believe this amounts to aesthetic appeal, as I don't believe any of us have ever 'seen' a real triangle by definition. What we have seen is what we want to believe.
It just seems like "contradicting" ourselves is only possible because we are are adamant language = truth and words must mean something in the strictest sense. Like since man has deemed it so...words must have 'truth'. This would be more of a contradiction, I would think...as what makes us so sure the definition to any word is ultimate 'truth.' It may be that the essence of language stops us from seeing truth....or at least our stanch ideals of the aesthetics of such language meaning some sort of unwavering 'truth.'
Why should we convince ourselves definitions like "a triangle has three sides" must be strictly adhered to or else we commit the 'sin' of contradiction?
I don't think that words would necessarily become meaningless if we venture into more specific definitions. As we could still use the word's definition as a generalization, while also knowing that a triangle has much more than just 'three sides' and not contradict ourselves on a deeper level. I think Fractals may prove this.
I just don't see the problem with being honest about a thing...it's not like the world will cave in if we are. There will still be the idea of a triangle.