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# Golden Powers

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### Gold Again

### Pythagorean Golden Means

### Golden Triangle

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions

You will probably need to experiment a little to try to form a conjecture for the form of the coefficients. You can then use induction to try to prove your conjecture.

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

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.

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.