Integer Triangles with Integer Areas

Let us define an Integer Triangle as a triangle with integer lengths for all three of its sides. Some, but no all, Integer Triangles also have an integer area. The classic 3-4-5 right triangle is a simple example. The area is 3 * 4 / 2 = 6. In fact it seems that a lot of Right Integer Triangles have an integer area. We also saw on our most recent test an example of a Non-Right Integer Triangle with an integer area: a 4-13-15 triangle is not right, and it has an area of 24. So, a couple of questions come to mind:

1) Can you find a Right Integer Triangle that does *not* have an integer area? If so, given an example; if not, give a proof of why it is impossible.

2) Can you find any additional Non-Right Integer Triangles, other than the somewhat obvious 8-26-30, etc.?