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About Pythagorean golden means
What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?
Golden ratio
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articleLeonardo of Pisa and the golden rectangle
Leonardo who?! Well, Leonardo is better known as Fibonacci and this article will tell you some of fascinating things about his famous sequence.
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articleThe golden ratio, Fibonacci numbers and continued fractions
An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
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articleWhirling Fibonacci squares
Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.
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problemPythagorean golden means
Age16 to 18Challenge level
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. -
problemDarts and kites
Age14 to 16Challenge level
Explore the geometry of these dart and kite shapes! -
problemGold again
Age16 to 18Challenge level
Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72. -
problemGolden triangle
Age16 to 18Challenge level
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. -
problemGolden powers
Age16 to 18Challenge level
You add 1 to the golden ratio to get its square. How do you find higher powers? -
problemPent
Age14 to 18Challenge level
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.