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Hippocrates’s First Theorem

Over the half term I was doing some reading for my MA and I happened across Hippocrates’s First Theorem. (Not THAT Hippocrates, THIS Hippocrates!)

Here is the mention in the book I was reading (Simmons 1993):

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It’s not a theorem I’d ever come across before, and it doesn’t seem to have any real applications, however it is still a nice theorem and it made me wonder why it worked, so I set about trying to prove it.

First I drew a diagram and assigned an arbitrary value to the hypotenuse of triangle A.

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I selected 2x, as I figured it would be easier than x later when looking at sectors.

I then decided to work out the area of half of A.

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A nice start – splitting A into two smaller right angled isosceles triangles made it nice and easy.

I then considered the area b. And that to find it I’d need to work out the area the book had shaded, I called this C.

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Then the area of B was just the area of a semi circle with the area of C subtracted from it:

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Which worked out as the area of the triangle (ie half the area of A) as required.

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This made me wonder if it worked for all triangles that are inscribed in semi circles this way – ie the areas of the semicircles on the short legs that fall outside the semicircle on the longest side equal the area of the triangle.

My first thought was that for all three vertices to sit on the edge of a semi circle in this was then the triangle must be right angled (via Thales’s Theorem).

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I called the length eg  (ie the diameter of the large semi circle and the hypotenuse of efg) x and used right angled triangle trigonometry to get expressions for the two shorter sides ef and fg. Then I found the area of the triangle:

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Then the semi circles:
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I then considered the diagram, to see where to go next:

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I could see that the shaded area needed to be found next, and that this was the area left when you subtract the triangle from the semicircle.
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I could now subtract this from the two semi circles to see if it did equal the triangle.

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Which it did. A lovely theorem that I enjoyed playing around with and proving.

I think there could be a use for this when discussing proof with classes, it’s obviously not on the curriculum, but it could add a nice bit of enrichment.

Have you come across the theorem before? Do you like it? Can you see a benefit of using it to enrich the curriculum?

Reference:

Simmons M, 1993, The Effective Teaching of Mathematics, Longman: Harlow

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  1. April 11, 2016 at 4:14 pm

    I believe that I ran across this in William Dunham’s Journey Through Genius, which was this lovely little book of ingenious mathematical proofs. The proof, if I’m not mistaken, appeared in the chapter on compass-and-straightedge proofs. (That’s the chapter which got me to finally understand how to turn the area of an arbitrary polygon into a square, the classic quadrature.)

    • April 11, 2016 at 4:20 pm

      Nice, I may have to find myself a copy of that!

  2. April 11, 2016 at 6:11 pm

    Numberphile did a lovely job with lunes. Worth a watch

    • April 11, 2016 at 6:15 pm

      Cheers, I will indeed have a watch!

  1. April 12, 2016 at 4:01 pm

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