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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

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
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 $x$.