a geometric AM-GM derivation
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- čas přidán 4. 04. 2023
- a geometric AM-GM derivation. We prove the arithmetic-geometric mean inequality by comparing areas of triangles. This generalizes to higher dimensions with boxes, although I'm not sure of the proof. AM GM says that the geometric mean sqrt ab is always less than or equal to the arithmetic mean a+b/2
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For the 3D case: Label some corner Z of the cuboid and the three adjacent corners A, B and C such that |AZ|³ = a, |BZ|³ = b and |CZ|³ = c. Then place the square pyramides such that the heights are exactly those AZ, BZ and CZ oriented with the tip beeing Z, i.e. the bases contain the points A, B and C respectively. Observe that on the face spanned by AZ and BZ the 2D situation appears; similarly on the other faces, so the pyramides have the required volumes and fit perfectly overlapping the cuboid. Observe that equality holds in the cube case. Also the same construction of fixing Z nicely works in any dimension.
Perfect
Could you explain it in an easier way. I need help visualizing the reorientation of the square pyramides
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AM-GM inequality can also be proved using Calc III (n-dimensional).
long time no see, Dr. Peyam. I hope you are well. Welcome back.
Another prove for 2d case uses triangle inscribed in a semicircle, maybe this can be generalized to bigger dimensions.
Dr peyam, you have to think in PIRAMIDS!!! the volume of a piramid base 3√a sqare and height 3√a will be a/3. There fore onde of the puramids necessarily will be greater or equal to the retangular block. Thus we prove the AMGM
Very nice!!!
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Cool!!!!
This is a mathematician's take on spring break...QED.
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Wouldn't a and b have to be greater than 1 for this to work?
You mean 0?
At 1:20 I learned a new congruence theorem AAA. Two triangles with the same angles are congruent. 😢 Dr Peyam is getting rusty!!
Isn’t that true? Congruent as in similar
@@drpeyam it may be a language difference. Congruent triangles are the same shape and size, so equal angles and equal sides. Similar triangles are the same shape, so just angles are equal, the sides don’t have to be equal.
So to me it sounded odd to hear the triangles are “congruent” therefore a = b?
The triangles were similar, therefore both are isosceles, (same shape) so the other side is also b.
Good to see you, et vous pouvez parler en français, c'est ma langue maternelle!
As opposed to an arithmetic derivation
Not convinced.
Ok