Hi Bjorn.

Let D be the depth of the shelf and let x be the length from the wall to the beginning of the cut.

Thus, (D - x) is the length of the leg of the isosceles right triangle you removed. And of course, the hypotenuse of that triangle is x, so that it matches the segment connecting to the wall.

So you have a right triangle with legs (D - x) and hypotenuse of x. Pythagoras tells us:

x² = 2(D - x)²
x² = 2(D² - 2Dx + x²)
x² = 2D² - 4Dx + 2x²
-x² + 4Dx = 2D²
x² - 4Dx = - 2D²

Now we can "complete the square" to determine the value of x:
x² - 4Dx + 4D² = - 2D² + 4D²
(x - 2D)² = 2D²
x - 2D = ±D√2
x = D(2 ± √2)

But we know that x < D, so D(2 + √2) doesn't make sense. Thus the answer is this:
x = D(2 - √2)

Let's test it. Say I have a 20cm deep shelf. Where should I cut that 45 degree angle?

x = 20(2 - √2) = 20(.5858) = 11.7 cm from the wall.

Doing so, I will have cut off a piece that is 20 - 11.7 = 8.3 cm long. As a right angled isosceles triangle, the hypotenuse would be √(2[8.3]²) cm.
√(2[8.3]²)
= √(2[68.89])
= √(137.78)
= 11.7 cm as expected.

Cheers,
Stephen La Rocque.