Heron's formula: the classic proof.

The classic proof of Heron's formula for the area of a triangle, using only Pythagoras's law and algebra.

[@tedszy7100]

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[@terrycymru9549]

I bet this guy is a lot of fun in the lecture room, I tost the will to live when he started with the proof

[@tedszy7100]

Don't give up, man! Life is worth living!

[@stevenmayhew3944]

I noticed that other KZbinrs use S for the Semiperimeter rather than p to avoid confusion.

[@tedszy7100]

In defense of "p", I will appeal to a higher authority: Soviet Mir books use "p"!

[@NadiehFan]

@@tedszy7100 That's not a matter of authority. In Russian _p_ is used since this is the Latin equivalent of the first Cyrillic letter of полупериметр _semiperimeter_ and similarly in some other Slavic languages, e.g. Polish, Ukrainian, Bulgarian, but _not_ in e.g. Slovenian, Croatian, Serbian, Macedonian. Also, in French, _p_ is used to avoid confusion with the use of the capital _S_ of _Surface_ for 'area'. But in most other languages including English _s_ is commonly used to designate the semiperimeter. The usage of _s_ for the semiperimeter of a triangle dates back to Leonhard Euler who gave an original geometric proof of Heron's formula and _s_ has been in common use ever since (like so many other notations introduced by Euler). So, you really should have used _s_ here. Note that Heron's original proof was only discovered in 1896 so this was unknown to Euler in the 18th century.

[@GopanNeyyar]

@@NadiehFan I prefer p solely for a different reason. It ends with a vowel sound 'ee'. So, it is easier pronounce when the next syllable is a consonant. So, 'p minus a' is easier to pronounce than 's minus a' (the letter 's' ends with a consonant sound, and the word 'minus' begins with a consonant sound. So it is difficult to pronounce them in succession). When you compute the surface area of a pipe, the term (h + t) appears : where h is the height and t is the thickness. It is easier to pronounce t + h rather than h + t for the same reason mentioned above. This is a more insisting example.

[@wayneyadams]

That is exactly the way I did it, I even used x in the left triangle and (a-x) in the right triangle. I teach Physics, not math, so I have never tried to prove this before, but I think it is important to prove every equation taught to students so equations don't become hand waving. I am now inspired to do it by cotangents. I have no idea what excircles are, so that's off the table.

[@tedszy7100]

I am glad to heat that you came up with the same proof independently. I agree that people should know the proofs of fundamental results like Heron's formula, AM-GM, Ceva's theorem etc., especially mathematics competitors. I will do a vid on the excircle proof soon.

[@user-ky5dy5hl4d]

Physicists are not mathematicians and the other way around. But if you are a physicists maybe you can help me with a couple of questions in physics; what causes the speed of light? What is the definition of time?

[@wayneyadams]

@@user-ky5dy5hl4d I beg to differ, I have a book on my shelf titled "Mathematics for Physicists" that is over 800 pages. ------------------------------------------- Light is the part of the electromagnetic spectrum that we see, so the question really is what determines the speed at which electromagnetic waves travel through empty space (vacuum) or any material for that matter? Magnetic fields are generated by moving electric charges or fluctuating electric fields. An oscillating electric field (a wave) generates a magnetic field perpendicular to itself. If you point the fingers of your right hand in the direction of the electric field at any moment and curl them toward the magnetic field at that moment your thumb points in the direction the wave travels. Here is a link to a very good image of what i am talking about. c8.alamy.com/comp/2AG6CWX/electromagnetic-wave-structure-and-parameters-vector-illustration-diagram-with-wavelength-amplitude-frequency-speed-and-wave-types-2AG6CWX.jpg Since the wave is part electric field and part magnetic field, we need to see how easily those fields can form, or put another way, what is the resistance of space to the formation of the respective fields. You can think of it as a kind of resistance or drag on the fields. We call the resistance to the electric field permittivity and it uses the Greek letter epsilon. The resistance to the magnetic field is called the permeability and it uses the Greek letter mu. The symbols are easy to remember, Greek e for electric field, Greek m for magnetic field. Together they determine the speed at which an electromagnetic wave can travel. The equation for the speed of light is: C = square root [1/(epsilon x mu)] When light enters a substance the values of epsilon and mu increase resulting in a lower speed. The greater those values the slower light travels through that material. If you want to know the real reason light travels slower through materials like water, here is a link to a great video that explains what is really happening. kzbin.info/www/bejne/eYbNpWZsiKlmnpo Here is the companion video that explains why light bends when entering or leaving water. kzbin.info/www/bejne/hH3QoYGDZdiaj80 -------------------------------------------- Defining time is more difficult. There are numerous definitions, some simple and some esoteric and difficult to understand, and a few that border on science fiction. The best thing I can recommend is that you do a KZbin search, watch a few videos and decide which makes the most sense to you. Alternately, you can do a Bing or Google search and read about the various theories. -------------------------------------------- If you have any further questions don't hesitate to ask. I would recommend you go to Quora and ask your questions. The reason is that i can upload diagrams and equations where that cannot be done here. Also, if you go to my page Wayne Adams can find several questions i have already answered. When you ask your question you have the option to direct it to one specific person so just direct it to Wayne Adams and I will get a notification. WWW.quora.com

[@MissPiggyM976]

Wonderful Heron's !

[@johnnyragadoo2414]

I put this on my watch later list because this is a challenge I want to solve for myself. I suspect my method - if it works at all - will be different from yours. Restating Heron's formula can yield (man, I wish youtube supported mathjax): triangle area = sqrt(2(a^2b^2 + a^2c^2 + b^2c^2) - (a^4+b^4+c^4)) / 4 Each of those terms with exponents could be thought of as a 4-d prism. The square root of a 4-d volume is a 2-d surface area, which is what we want to solve for. That makes me wonder if there is a "straightedge and compass" proof of Heron's, borrowing a little real estate from a fourth spatial dimension. Yeah, I know, I sound like Terrence Howard when I say that - but, there's the equation. It works. It is equal to Heron's in its traditional form, and it's got terms to the fourth power. The 2/4 power (square root) of a 4-d volume is the area of one side, just like the 2/3 power of a 3-d cube's volume is the surface area of one of its sides. I'm either intrigued or just very easy to amuse.

[@tedszy7100]

Sometimes, having recourse to the 4th dimension can make things happen in geometry. An example of this is quaternions.

[@gavintillman1884]

s is the standard symbol for semiperimeter

[@bpark10001]

How do you cover the case when angle B or C is obtuse? That puts the h line from A exterior to the triangle, requiring extending line BC. Yes you could "turn" the triangle so A is the obtuse angle, but that is a weak argument. x would become negative?

[@guyhoghton399]

I assume you would then drop the perpendicular from the obtuse angle to the opposite side, so the _h_ line would still be internal to the triangle. Just relabel the vertices if you want to use the professor's proof verbatim.

[@bpark10001]

@@guyhoghton399 Yes but if you do that, you need to place limitations upon the side lengths at the start. This destroys the beautiful symmetry of the problem. On the other hand, if you allowed negative length, the area of one of the triangles would be "automatically" subtracted (as it should), instead of being added, without any special handling of the math because of the negative-length base, & the symmetry would be preserved.

[@SteveMA1000]

Your perpendicular needs to stay inside the triangle. Pick the vertex of the obtuse angle to keep it inside. Procedure is the same from there on.

[@DrTinkerJim]

Very clear presentation ! Thank you....

[@tedszy7100]

You are welcome!

[@perfredlund]

This proof does NOT consider all the different possible shapes of a triangle. So there are more cases than this actually!! 😁

[@primkiddee]

Thanks for Heron's formula 💕

[@tedszy7100]

You are welcome!

[@NadiehFan]

This is _not_ Heron's original proof, which relies only on similar triangles. See Heath, _A History of Greek Mathematics, Volume II_ p. 321-323 for Heron's original proof.

[@godfreypigott]

Why does it have to be?

[@NadiehFan]

@@godfreypigott It doesn't have to be. Tedsky mentions a number of known proofs but fails to mention Heron's own proof and then proceeds to explain what he calls the _classical proof_ which is at the top of his list of proofs. This may convey the idea that the proof as presented in this video is Heron's own proof, which it is not.

[@godfreypigott]

@@NadiehFan Classical does not mean original, and I am quite sure no one thought that.

[@Gunslinger-us1ek]

didnt know it was THAT easy, tysm

[@tedszy7100]

It is indeed easy, but only when broken down into easy-to-grasp steps.

[@QuaDue]

we used s in the school, s for semiperimeter, how weired is that ;)

[@tedszy7100]

I admit that "s" is a pretty reasonable choice for semiperimeter.

[@islandfireballkill]

What application are you using to draw? It looks good.

[@tedszy7100]

I used Asymptote to draw the figures.