Two Tilted Rectangles
I found this problem from the wonderful Math with bad Drawings site in my news feed. They cite it from Carolina Shearer. Her twitter account contains more such nice problems. It is the kind of problem which seems only adequate for advanced students. Sometimes, you can solve them by looking at it in the right way. Most of the time, you will start a lengthy computation. Often, you will notice in retrospect that the solution was quite easy and you could have guessed it.
For this problem, I did not see the solution. The problem, of course, is to fit two equal rectangles into a square in the shown manner. For a start, try to find a reason why the point 1/2 on the lower side solves the problem. You can use the general fact that a=b+c in the figure below.
I admit that I did the computations using Euler Math Toolbox. That works. However, there is an elementary fact that can be used to prove a+b=c. For a hint, consult the following image. If the green lines are equal the red lines are equal.