Heron’s formula: What is the hidden meaning of 1 + 2 + 3 = 1 x 2 x 3 ?

[Music] welcome to another mythology video let me start by showing you a very surprising connection between two mathematical gems first up we have 3 squared plus 4 squared equals 5 squared and it's associated right angle triangle you've seen that one a million times right the second gem is this 1 plus 2 plus 3 equals 1 times 2 times 3 well maybe not so much a gem as something that is usually passed off as a fun curiosity anyway what have the two in common can you guess well it turns out a lot to start with one plus two plus three is equal to six and six is also the area of the triangle let's check area with a triangle that's base times height divided by 2 and so base 4 times height 3 that's 12 divided by 2 equals 6 correct but that's just a start have a look whoa what just happened well what we've thrown in here just touching the three sides of the triangle is called the in circle and as indicated the three touching points split the sides into segments of lengths one two and three one plus two is three one plus three is four and two plus three is five okay but all this could still be passed off as some sort of small number coincidence right but there's more have a look at the square down there since all sides of a square are the same the radius of the in circle must also be one right that's skewed and it's probably also not something you've ever noticed in your previous million encounters with the mega famous 345 triangle but the truly mind-bogglingly amazing thing is that everything you see in front of you is true for every triangle with in circle radius one there the area of a triangle like this is always equal to the sum r plus g plus p which is always equal to the product r times g times p here always two more nice specific examples of this miracle in action that's the special case of an equilateral triangle root 3 plus root 3 plus root 3 is equal to root 3 times root 3 times 3 is equal to the area of the triangle pretty and very easy to check and how about this isosceles triangle which features the golden ratio phi very nice and also not terribly hard to check that this really works maybe one of you can do the checking in the comments now it also turns out that what you see in front of you is the heart of a very famous mathematical formula heron's formula heard of it well the plan for today is to rediscover visualize and prove heron's 2000 year old formula as well as an amazing 1 400 year old extension due to the indian mathematician brahma gupta you may have heard people go on about how beautiful and accessible great maths can be well let me show you ready here we go [Music] first up let's reconstruct heron's formula from its heart okay let's tidy up what we've got r times g times p that's just rgp the perimeter of the triangle consists of two r's two g's and two p's so the sum r plus g plus p is equal to half the perimeter and so let's abbreviate the sum by the letter s which is supposed to simultaneously stand for sum and semi perimeter half the perimeter slicker summarizing two birds with one stone all right i'll stop now so far well see i stopped so far it's been all about triangles is in circle radius one to get the corresponding two equations for triangles with general in circle radius little r we just have to scale everything inside by little r here when it comes to the formulas we have to take into account that area is a 2d quantity that the sum of three lengths is a 1d quantity and that the product of three lengths is a 3d quantity ready for the magic playing captain picard now engage cool actually nothing has really happened yet we are still looking at the same heart as before we've just spelled out what this heart looks like with a little bit of scaling thrown in now we've got two equations both containing little r to get that famous formula we just have to eliminate a little r let's do this with some mathologer algebra autopilot there multiply through is s okay i think you know where we're going with this right okay and there that's heron's famous formula for the area of the triangle well those among you who are familiar with herons formula will complain no that's not heron's formula and those of you who've never seen the formula before will most likely be a little bit underwhelmed because this formula does not appear to be very useful right after all faced with a triangle in the wild how are you supposed to figure out what those four factors rdp and s in the formula are okay so maybe a semi-interesting formula sorry bad habit well on closer inspection the factors in our formula are just the length of the very accessible sides of the triangle just slightly disguised wait what yes let me show you first s that's just the sum of r g and p but remember it's also half the perimeter but half the perimeter is also just a plus b plus c divided by two right cool s can be expressed in terms of the three sides what about r well notice that g plus p is equal to a and so r is just s minus a and similarly for g and p there and there okay square root on both sides there that's the area of the triangle expressed in terms of the three sides of the triangle that's heron's formula as it's usually written cool and now a little break for a history lesson who was heron heron alexandria as he is usually called was a greek mathematician and engineer who lived from about 1080 to 70 a.d and who's been called the greatest experimentalist of antiquity good guy in terms of mathematics it's this formula that he's mostly remembered for although there is some debate over whether it was really heron who first discovered it sadly heron's formula just like a lot of other beautiful mathematics is pretty much extinct in australian schools how's the situation in other parts of the world were you taught heron's formula in school to what extent is it still taught where you live should it be taught let us know what you think in the comments anyway if we are not worried about using a bit more virtual ink we can also get rid of the s by writing the formula entirely in terms of a b and c doing this and simplifying a bit gives this version of the formula also very nice and symmetrical i actually prefer this one now speaking of symmetry have a close look at those factors under the square root sign in some way despite its symmetry our formula is still a little bit lopsided with the last factor featuring all plus signs unlike the other three factors which also feature one minus sign each can you see it well we can't have that can we so let's hammer some more symmetry into the thing we'll do this by including a fourth side d of length zero yep a fourth zero side of our triangle and we puzzled why people think mathematicians are weird anyway we'll subtract the zero length d from the last factor and add it to the first three factors interesting right there minus a plus the remaining stuff second factor minus b plus the remaining stuff all factors now have the same structure very pretty and something that pretty must have some real meaning even if d is not zero right a b c d four sides well maybe that's the formula for the area of a quadrilateral that'd be very nice but obviously can't be that simple right because there are always lots of different quadrilaterals with the same length sides and different areas so what could be going on here well around 620 a.d the famous indian scientist and mathematician brahma gupta figured out that this really is the area of a quadrilateral as long as it is cyclic that is as long as the four vertices of the quadrilateral are contained in the perimeter of a circle very beautiful but also very weird not least the position of the circle right we saw a circle earlier and so if a circle is involved with a quadrilateral i'd have guessed that it's an in circle or something touching all sides from the inside but no it's from the outside like this anyway the quadrilateral formula definitely agrees with what we've done as you shrink d to zero the quadrilateral turns into a triangle and brahmagupta's formula turns into heron's formula cool actually brahmagupta's formula looks even cooler if you write it in terms of the semi-perimeter of the quadrilateral like that whoa definitely another contender for the most beautiful formula in mathematics don't you think but even that is not the end of the story it turns out that there's actually a general formula for the area of any quadrilateral this formula was discovered in 1842 by the mathematician carl anton bretschneider in fact bretschneider's formula extends brahmaguptas by including an additional correction term under the square root sign this correction term depends upon a pair of opposing angles in the quadrilateral ready for this so red schneider just throws in a little trick and gets the formula for the area of any quadrilateral pretty spectacular and of course if the universe makes sense right then brechnider's correction term must vanish for cyclic quadrilaterals and it does and that's also easy to see the key is the famous characterizing property of cyclic quadrilaterals that opposing angles always add to 180 degrees not sure whether you've heard of this or not anyway it's true and so alpha plus beta is equal to 180 degrees in cyclic quadrilaterals and that means that alpha plus beta divided by 2 is 90 degrees and of course the cosine of 90 degrees is zero beautiful stuff don't you agree now before we get to the pretty proofs in the next chapter here are a few more questions for you to ponder and answer in the comments first of all a trivia question why did i go for labeling rgp red green purple instead of the more obvious rgb red green blue second why was marty campaigning for yellow green purple third bretschneider's formula even works for quadrilaterals that are not convex like this one but what happens if the quadrilateral crosses over itself what is the measure then final hardcore challenge can you prove that the correction term can also be expressed in this beautiful non-trig way well let us know in the comments [Music] okay so how do you discover something like heron's formula and how do we prove that beautiful 1 plus 2 plus 3 equals 1 times 2 times 3 geometric heart is really there well this is where it gets really interesting for people like me and i hope for you too so let me show you well what we're interested in are areas of triangles so everything must start with the formula for the area one half base times height and quickly where does that formula come from well that can be seen at a glance right just double the triangle split the clone and flip over the pieces there this gives a rectangle with area base times height and so the area of the triangle is half the area of the rectangle half base times height easy now on to discovering heron's formula a triangle is completely nailed down by its three sides and so it's definitely very natural to look for a formula for the area in terms of the length of these three sides right finding such a formula is actually not that hard and anybody who's played around a bit with trigonometry in school should be able to invent such a formula for themselves just quickly and here i give you a license to space out for the next couple of seconds if you hate trig anyway just quickly over there the base of the triangle is capital b and the height we express in terms of the angle between the sides a and b remember you're so katoa so opposite over hypotenuse and yep that height is just a times the sine of the angle almost there we want to get rid of that trig bit the sine of the angle and that's also pretty easy because the cosine of this angle can be written in terms of the sides a b and c using the cosine formula like this but of course pythagoras for trick tells us that sine squared plus cos squared equals one and so at the cost of a little square root mass we can express the sign in terms of the cosine plug in and we're done too fast don't worry i promise you this will not be on the test all i really want you to take away from this part is that getting an area formula in terms of the size is actually not that hard and in fact our messy formula up there is essentially heron's formula a few algebraic manipulations and we get the formula in its familiar form feel free to give it a go in the comments again not terribly terribly hard if you know in advance what you're aiming for on the other hand if you don't know heron's very symmetric formula you'll most likely never find it or it's beautiful heart if you've derived your abc formula in this natural but no-brainer manner but hardcore mythologies want better than that right we want a natural and beautiful way to discover heron's formula and its heart and yes of course i've got one for you want to see it well of course you do again very beautiful mats ahead so enjoy now this other natural way involves splitting the triangle into three small triangles whose spaces are a b and c here's a quick animation [Music] there the area with triangle in terms of its three sides and its in circle radius little r easy and of course a plus b plus c divided by 2 is just our semi-parameter s and do you remember areas equal to s times r was also the first of our two equations from earlier on neat that was quick but not quite what we want because our area formula also includes that little r so what would a modern heron do to get rid of that little r well for that it is natural to look for a second way to interpret a diagram in front of us and there is one obvious one that jumps out the dissection of the triangle into six little right triangles there ah six right angled triangles and in this way we have very naturally arrived in rgp territory and now we can hunt for our second equation the sum equals product equation so where is the second equation hiding well let's see what have we got here six right angle triangles with any two of the same color being identical okay what about the non-right angles in these triangles let's move some triangles out of the way so we can see now what's the sum of the three marked angles the thick angles in the middle can you see well all the way around that's 260 degrees of course but then the three marked angles jump to just half that that's 180 degrees got it now what about those thin angles at the corners of our original triangle well the angle sum in the triangle is 180 degrees and half that is 90 degrees okay thin angles add to 90 degrees there nice and thick angles add to 180 degrees ooh also nice but looks like we're just doing that very badly at 10 grams doesn't it well next in our journey of discovery it's natural to shuffle the left pieces and the right pieces around a bit to make the diagrams look as similar as possible let's go for this okay shuffle shuffle shuffle shuffle okay yep looks quite similar now but what else well what about if we rescale a bit like this like that and like that now that looks really promising actually pretty obvious now that things will fit together like this right scaled versions of two green triangles a red triangle and a purple triangle fit together perfectly into a rectangle fantastic there should be an equation in there let's go and find it for this we have to go back and figure out how exactly the original triangles have to be scaled to fit together okay so let's start with four of the original triangles a red a purple and two greens [Music] we want to scale the labeled green and red triangles so that the two highlighted sides become equally long for that let's give a name to the unlabeled side let's call it h so to make the two highlighted sides fit we scale the green triangle by r and the red one by h here we go there and there perfect fit okay to fit the remaining green triangle we just have to scale it by little r there fits again nice okay just sorting out a couple more things here as we go along all right okay now as usual we want to rescale to make the two highlight sides fit well that will be the case if instead we make the two horizontal sides fit clear okay so scale the trapezium shape at the top by p and the purple triangle by r plus g times little r okay okay okay now can you see the second equation we've been looking for no ready for a really nice aha moment well opposite sides of a rectangle of equal lengths right and so ta-da that's our second equation the sum equals product equation and what i just showed you is the most natural simple and accessible way to discover the heart of herron's fantastic symmetrical formula that i can think of now the first time i saw the point being made that what we see here is really the heart of herron's formula was in a quora pose by job baumann and i learned about the slick scaling way of finding the sum equals product identity from the amazing cut and not side this website is a home of around a thousand interactive animated mathematical gems in the case of hair on this is what the interactive part looks like very slick huh quick quiz for you what does the rectangular bit at the bottom illustrate oh and here's a bit of good news most of the animations on the cut the knot side were java applets and once java was no longer supported by internet browsers and these applets stopped working the site lost much of its appeal a real shame but now since quite recently a new very nice chrome extension makes it possible to run the defunct java applets on the cut the knot side and other sides again absolutely made my day when i found out about this check it out i'll put a link to the chrome extension in the description of this video to finish off i've animated another absolute gem of a proof for you the 2012 super simple one page derivation of baragupta's formula from herron's formula by the mathematician albrecht hess that this proof was missed for hundreds of years is absolutely incredible the main ingredient of the proof is that in general the area of a cyclic quadrilateral is really just the difference of the areas of two similar triangles those two nested triangles in the picture over there are actually similar then you simply express the two areas using heron's formula follow your nose and as if by magic brahmagupta's formula materializes well enjoy the animation of the details of this proof and the music and that's all for today until next time [Music] [Music] [Music] so [Music] [Music] [Music] [Music] [Music] [Music] so [Music] [Music] [Music] so [Music] [Music] so [Music] you

Channel: Mathologer

Views: 504,449

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Id: IguNXoCjBEk

Length: 26min 40sec (1600 seconds)

Published: Sat May 14 2022