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Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Golden Triangle

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

Golden Ratio

Age 16 to 18 Challenge Level:

Why do this problem?

This problem introduces the Golden Ratio as the solution of a quadratic equation and links to many other investigations. A solution to the equation can be found by trial and error.

Possible approach?

To motivate the topic why not introduce several problems where the students can discover that the Golden Ratio occurs in very different contexts and reinforce their own understanding of the algebra that occurs. For example:
Golden Powers , Golden Triangle and Golden Eggs.

Key questions

If $\phi $ is the golden ratio then what is $\phi^2$ and $1 + {1\over \phi}$ and does this suggest a way to simplify the final equation?