Cutting Down The Middle

Imagine drawing lines from (0,0) to (1,0), (1,1), (1,2), (1,3),... etc. Then from point (0.0) it's going to look like a row of telephone poles bunching up together into the distance. Which pair are furthest apart?

Let the right isosceles triangle in question have side length $s$ . With the pictured median drawn in place the following equations are known:

$x + y = \frac{\pi}{4}$ ;

$tan(y) = \frac{s/2}{s} = \frac{1}{2}$ .

This yields $y = arctan(\frac{1}{2}) \approx 26.57$ degrees, which in turn produces $x = 18.43$ degrees. Thus, $y > x$ .