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Exercise 8.10 at this site http://www.euclidea.xyz/ claims that harmonic mean can be built with just 4 simple objects (lines and/or circles)

My best result is 6 objects (including perpendiculars and segment bisector). Do you have any hint to simplify the construction?

As the page is not accessible if you have not done all the previous ones, here is a screenshot of the page screenshot

Raffaele
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  • Do you know how to construct a/b and ab given a and b? – user2277550 Oct 14 '16 at 13:53
  • Yes, but they involve Thales with parallel lines which "cost" many elementary elements. If I put (a+b) and b on a – Raffaele Oct 14 '16 at 15:17
  • Yes, but they involve Thales with parallel lines which "cost" many elementary elements. I draw two rays and I put (a+b) and b on one ray and (2a) on the other I am done. But this costs 6L and 13E, while the author wants 4L and 4E. (L are lines or circles and E are elementary constructions like parallels or segment transport) – Raffaele Oct 14 '16 at 15:23

2 Answers2

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$$H_\left(a,b\right) = \frac{2ab}{a+b} \Rightarrow \frac{H}{b} = \frac{2a}{a+b}$$

Mildly obfuscated solution illustrated here: Mildly obfuscated solution illustrated here.

As you can see, this solution will get you a 4L/7E score. Getting 4/4 is a little tricker. :D

Eman Yalpsid
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madbadger2742
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Construct a isosceles triangle with base $b$ and the other two sides as $a+b$ each. Draw a circle of length a from the top vertex. Join the two points where it intersects the triangle. That length is half the harmonic mean. Double that length with a simple compass operation. Use similarity to prove the result.

user2277550
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