Rectangle triangle problem

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September 1997
Triangle in rectangle

If we label the distances AP, PB, CQ and QB as shown above then we can write three equations for the areas of the triangles as follows:


Notice that the two ratios are the same! If we call this ratio r then we can calculate a polynomial for r as follows:


Taking the positive root gives us the golden ratio:



For explaining a solution like this one that uses very few variables, it's much easier to read if the variables are named with different letters, but no subscripts.

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