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.

@MarceloGondaStangler

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Perfect

@faizan2031

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Could you explain it in an easier way. I need help visualizing the reorientation of the square pyramides

Dr payem is back 🎉🎉🎉

Wow Dr.Peyam it's has been three months since your last video. So nice to see you back.

@drpeyam

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Thank you so much 🙂

Yayyyy welcome back!

Glad you're back was worried . Yay another neat beauty nice one 😃😃

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

@drpeyam

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Very nice!!!

Cool!!!!

This is a mathematician's take on spring break...QED.

Yee 🎉 Dr Peyam, Dr peyam 🤭 Sometime, i want you make a course with some grade ❤😊🤭

@drpeyam

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Omg thanks for being a member 😍😍

@ekadria-bo4962

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@@drpeyam Yea Dr Peyam 🫠 Sometime i want you to make some course.. 😅

Wouldn't a and b have to be greater than 1 for this to work?

@sujitsivadanam

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You mean 0?

Good to see you, et vous pouvez parler en français, c'est ma langue maternelle!

At 1:20 I learned a new congruence theorem AAA. Two triangles with the same angles are congruent. 😢 Dr Peyam is getting rusty!!

@drpeyam

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Isn’t that true? Congruent as in similar

@bobh6728

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@@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.

If I mail you an ASU shirt will you wear it in a video

As opposed to an arithmetic derivation

Not convinced.

@drpeyam

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Ok