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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.

Golden Eggs

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

(1) You will need to know, or find out, the formula for the area of an ellipse. All you have to do then is solve a quadratic equation.

(2) What happens if you square this strange expression?