The problem: unit square, ABCD. Extend AB as a Ray and trace a Ray from D, crossing the side BC at E and the Ray AB at F. EF = 1. What is the length of BE? (Image attached to scale)
This problem is "easy" to solve:
Similarity of BEF and CED. The problem is: you get a quartic equation. Looking at the valid solution for this context:
$$x = \frac{1}{2}\left( 1 - \sqrt{2} + \sqrt{2 \sqrt2 -1} \right) $$
Since this solution doesn't have any cubic nor quartic roots, it is constructable. The image attached was created with geogebra, to scale, using compass and straight edge contructions. The nested roots was made with a right triangle and a geometric mean.
I've been trying to find a geometrical solution for this problem which would avoid the quartic without success.
I also remember that the solution that I've seen uses trigonometry and arc sum (for a tagent, if my memory serves me right), but wasn't able to prove it.
Any inputs on this would be appreciated.
Source: I've seen this problem around 10 years ago, but I never knew it's source. If anyone recognises it, let me know and I'll update this post.