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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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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?

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Classical Means

Use the diagram to investigate the classical Pythagorean means.

Mean Geometrically

Stage: 5 Challenge Level: Challenge Level:1

diagram
This involves nothing more than areas of right angled triangles, using the symmetry in the diagram, and sines, cos's and tan's.

Well done M.S. Ezzeri Esa from Cambridge Tutors College, Croydon and thank you for this solution.

Let radius = $r$; $\angle AOD = \angle BOD = \alpha$
Area $ADBO$ = $2 ({1\over 2 }r^2 \sin \alpha) = r^2 \sin \alpha$
Area $ABO$ = ${1\over 2}r^2 \sin 2\alpha = r^2 \sin \alpha \cos \alpha$
Area $ACBO$ = $2({1\over 2}r^2 \tan \alpha) = r^2 \tan \alpha$
(Area $ABO$). (Area $ACBO$) = $r^2 \sin \alpha \cos \alpha\ .\ r^2 \tan \alpha = r^4 \sin^2 \alpha ={\rm (Area ADBO)}^2.$

The area of $ADBO$ is the geometric mean of the areas of $ABO$ and $ACBO$