How was this geometry problem created? 
This is a standard High School Olympiad problem and for an experienced problem solver a quite easy solve. But how was this problem created. To pose a problem, I believe is much harder, than to solve a posed problem. 
Here the problem poser had to first make the figure up and then simultaneously realise that $ND$ had the wonderful property of being equal in magnitude to the circumradius. Is there a nifty way to find out these wonderful geometric properties? 

This is NOT an answer to your question but is a shorter alternative version to the provided solution.
It should be clear that why the square-marked angles are right angles. Then, we can say that FNBD, and FBDO are cyclic quadrilaterals.

This means B, D, O, F, and N are all on the red dotted circle. Then, $\angle BNO = 90^0$. 
Result follows from the fact that DN = OB because they are the diagonals of the rectangle BDON. 
