A recent article in The College Mathematics Journal entitled The Shad-Fack Transom by Annalisa Crannell explores several methods of finding the radius of the small circle tucked up in the corner of a square circumscribed about another larger circle.
In one example Dr. Crannell found the radius expressed as the ratio of two radical expressions. She then “simplified” the expression because it was “a bit ugly” and asked whether new insights could be gained from the new form of the answer. This is a great question. In my classes I like to show students that the model (algebraic equation) that they have constructed for a word problem actually contains a description of the conditions of the problem.
Anyway here is my take on her challenge to find more solutions.
Consider this diagram where the large circle has radius one and the small circle has radius . The figure outlined in yellow is similar to the figure outlined in turquoise. By the property of similar objects all corresponding linear measurements are in the same ratio. Therefore the ratio of the radii of the circles to equals the ratio of the pink line to the green, that is, .
The expression can be simplified by multiplying the numerator and denominator by . The new expression for is . It is easy to see where this expression comes from if a few ancillary lines are drawn. In this drawing all the line segments () indicated by the blue arrows are equal and equal to . Since , which simplifies to .
I am beginning to think that is a beautiful number.