Which AI Technique best suited to discovering non-linear relationships in data? I am interested in exploring whether AI techniques can derive hidden patterns of relationships in a data set.  For example, from among house size, lot size, age of house and asking price,   what formula best predicts selling price?  

In explorations around how this might be done, I tried to use a neural network to solve for a predictable relationship between two variables to predict a third, so I trained my neural network with inputs consisting of the length of two sides of a triangle, and the result being the length of the hypotenuse.  It couldn't get it to work.      

I was told by somebody who understands all this better than me that the reason it failed is because conventional neural networks are not good at modeling non-linear relationships.  

If that is true, I wonder if there is some other AI technique that could 'derive' a network modeling the phythagorean theorem from a training data set with better results than a normal neural network?
You are mixing up lots of things here. Specifically, you seem to be lacking a basic understanding of artificial neural networks and what they can do (e.g. what type of articifial neural networks are linear classifiers/regressors and which can model non-linear relationships). 

Therefore, I'd take a step back and start with understanding the basics of AI. The go-to book for that is 'Artificial Intelligence: A Modern Approach' by Russel and Norvig. It might be a slower (and more theoretical) start but IMO that is the right approach to actually understand what you are doing.