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Gold Again

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

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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.

Gold Yet Again

Age 16 to 18 Challenge Level:
In the equilateral triangle, draw in the altitudes of the triangle and taking the radius of the circle as $1$ unit, calculate the lengths.

In the square it is easiest to take the side of the square as $1$ unit and then calculate the radius of the circle.

In the pentagon it is easiest to take the side of the pentagon as $1$ unit, and the chord length $XZ$ as $x$ units and then use properties of similar triangles. Derive a quadratic equation which you can solve to find $x$.