Question 37916
Probably the easiest way to see this is to draw the right isosceles triangle with the right angle at the origin of your coordinate axes. Then draw the horizontal leg a distance of a-units along the positive x-axis and the vertical leg a distance of a-units along the positive y-axis. Connect the ends of the x-axis leg and the y-axis leg to form the hypotenuse.
You can label the triangle with A at the right-angle vertex, B at the end of the x-axis leg, and C at the end of the y-axis leg.
With this labeling, the hypotenuse is segment CB, the x-axis leg is AB, and the y-axis leg is AC.

Now, place a point (E) on the x-axis leg at a distance of a/2 from the origin (this point bisects the segment AB) and another point (F) on the y-axis leg at a distance of a/2 from the origin (this point bisects the segment AC), then connect these two points.

The proof:
Using the fact that parallel lines have identical slopes, find the slope of the hypotenuse. Slope is rise over run and for the hypotenuse, the rise is distance a and the run is distance a so the slope is a/a = 1 (it's really a negative slope).
The slope of the line connecting the midpoints of the two legs is (a/2)/(a/2) = 1 (again, this is really a negative slope)
Since the slopes of the two lines are equal, the lines are parallel. QED