Kant and The analytic synthetic distinction

There are two types of propositions, those that are analytic and those that are synthetic. An analytic proposition is constructed by analyzing a concept, while a synthetic proposition is constructed by synthesizing (or putting together) concepts. Analytic propositions do not add any new knowledge, that is, the predicate is contained conceptually in the subject. Synthetic propositions amplify knowledge, that is, the predicate contains something not thought of in the subject. Kant takes this distinction a step further. While it seems obvious that a posteriori knowledge (knowledge from sense experience) is synthetic, and that a priori knowledge (knowledge independent of experience) is analytic, Kant introduces the a priori synthetic. Here are some examples of the three types of propositions:

a priori analytic: "Triangles have three sides"
- Here the predicate " have three sides" is part of the definition of what a triangle is: a three angled, three sided, closed planar figure.

a priori synthetic: "The sum of the interior angles of a triangle is equal to 180 degrees"
- this fact concerning triangles can be proven apart from sense experience. We can derive this fact simply by what we know about a triangle by its definition.

a posteriori synthetic: "The sides of the pyramids at Giza are triangular"
- This is a contingent (as opposed to necessary, as all true a priori propositions are) fact about the world, and it requires sense experience in order to be discovered.