The only thing left to prove, at least in my view, is JeremiahSmith's assumption that the right angle is indeed the angle between the two equal sides. I will proceed by contradiction.
First, let us assume that we have an isosceles right triangle ABC, in which sides a and b are equal, and angle B is right. Because opposite angles of equal sides are equal, this means that angle A is also a right angle. Now, as a result of Euclid's fifth postulate, we knwo that the sum of the angles in any triangle is 180°. This means A + B + C = 180, or A + 90 + 90 = 180, which means A = 0. The definition of a triangle states that the points A, B, and C are noncolinear. But if A = 0°, then the points must be colinear, and thus the assumption that ABC is a triangle is violated. Therefore, by contradiction, such a triangle cannot exist in Euclidean geometry. QED.
Now, to describe better what Vorn was talking about, there are fields of geometry in which people decided that Euclid was smoking crack when he came up with his fifth postulate. One such field, spherical geometry, says that the sum of the angles in a triangle can (and indeed always are) greater than 180°. In this geometry, such a triangle can, and does exist.
For a simple example, try a sphere with circumference 4. Hold it out in front of you. Now, place three points on this sphere: One at the topmost point, one at the leftmost point, and one at the point closest to you. Join these three points, and you get, in fact, an equilateral right triangle, with three sides of length 1, and three angles of 90°.
The isosceles right triangle Ogredude is talking about can, in fact, be created on any globe with two points anywhere on the equator and the third point at one of the poles. Notice that the right angle(s) in this triangle is not between the two equal sides, which is how you get around the argument that sqrt(2) is not rational.
