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Gold Again

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Pythagorean Golden Means

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.

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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 Construction

Age 16 to 18 Challenge Level:

Once you have done the construction the different parts of this investigation can be taken separately and there is no need to tackle every part.

The problem solver is led through several different ways of finding the value of the Golden Ratio $\phi$ and proving that the various relationships are equivalent.

Younger problem solvers can simply stop at measuring the lengths and finding that the ratio of the sides of the rectangle $AEFD$ is the same as the ratio of the sides of the smaller rectangle $CBEF$ you get by removing a square from $AEFD$. This shows that $AEFD$ and $CBEF$ are golden rectangles.

You need Pythagoras' theorem and to be able to work with surds to get an exact value of this ratio in terms of $\sqrt 5$.

Alternatively, if you call this ratio $\phi$, then you can deduce that $$\phi = 1 + {1\over \phi}$$ and use a graphical method to solve this equation.

Lastly, if you know about quadratic equations, you can solve the quadratic equation to find the value of $\phi$.