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

Today's video is about Heron's famous formula and Brahmagupta's and Bretschneider's extensions of this formula and what these formulas have to do with that curious identity 1+2+3=1x2x3. 00:00 Intro 01:01 1+2+3=1x2x3 in action 02:11 Equilateral triangle 02:30 Golden triangle 03:09 Chapter 1: Heron 06:18 Heron's formula 08:50 Brahmagupta's formula 10:20 Bretschneider's formula 11:52 Chapter 2: How? The proof 12:57 Heron via trig 20:09 Cut-the-knot 21:16 Albrecht Hess 21:46 Heron to Brahmagupta proof animation 25:10 Thank you! Heron's formula on the Cut-the-knot site: https://www.cut-the-knot.org/Curriculum/Geometry/HeronsFormula.shtml Original article by Roger B. Nelsen "Heron's formula via proofs without words", featuring a version of the nice rectangle proof that I am presenting in this video: https://www.maa.org/sites/default/files/0746834212944.di020798.02p0691h.pdf Simple derivation of Heron's formula just using Pythagoras's theorem: https/www.mathpages.com/home/kmath196/kmath196 Job Bouwman's maths posts on Quora (you'll have to scroll a bit to get to Heron's formula) http://shorturl.at/gzGOX http://shorturl.at/dBX12 https://www.quora.com/profile/Job-Bouwman A very comprehensive book about quadrilaterals: Claudia Alsina, Roger B. Nelsen - A Cornucopia of Quadrilaterals (Dolciani Mathematical Expositions) (2020, American Mathematical Society) Albrecht Hess's paper "A Highway from Heron to Brahmagupta" https://forumgeom.fau.edu/FG2012volume12/FG201215.pdf If you liked the rectangle proof of the sum = product identity you'll probably also like this proof of Pythagoras's theorem: https://youtu.be/r4gOlttnJ_E I also mentioned this one earlier in a video on this main channel https://youtu.be/r4gOlttnJ_E Two more interesting notes on the cut-the-knot page: 1. Let the angles of the triangle be 2α, 2β, 2γ so that α + β + γ = 90°. The identity RGP = r²(R + G + P) is equivalent to the following trigonometric formula: cotα + cotβ + cotγ = cotα cotβ cotγ, where "cot" denotes the standard cotangent function. More on this here https://tinyurl.com/yrsuhthk 2. A supercute way to derive Pythagoras from Heron with one line of calculus https://www.cut-the-knot.org/pythagoras/HeronsDerivative.shtml For a cyclic quadrilateral that also has an incircle we have a+b=c+d and it follows that the area is just square root of the product of all of the sides. A 3d counterpart to Heron's formula: https://en.wikipedia.org/wiki/Heron%27s_formula#Heron-type_formula_for_the_volume_of_a_tetrahedron A different 3d connection (de Gua's theorem) https://www.mathpages.com/home/kmath226/kmath226.htm A couple of links to get you started on generalisations involving cyclic n-gons: https://arxiv.org/pdf/1203.3438.pdf https://arxiv.org/pdf/1910.08396.pdf https://tinyurl.com/tyhzwpxj Another interesting observation extending the fact that the 3-4-5 right-angled triangle has incircle radius 1: In general, the incircle radius of any right-angled triangle with integer sides is an integer. Have a look at this for a related proof that arctan 1 + arctan 2 + arctan 3 = pi: https://www.geogebra.org/m/A65eMkuN https://math.stackexchange.com/questions/197393/why-does-tan-11-tan-12-tan-13-pi (2nd proof) Not many integer solutions for x+y+z=xyz: 0+0+0=0x0x0 1+2+3=1x2x3 (-1)+(-2)+(-3)=(-1)x(-2)x(-3) Other interesting little curiosities (some mentioned in the comments): 2+2=2x2=2^2 (of course) 3^3+4^3+5^3=6^3 = 6*6*6=216 illuminati confirmed 6+9+6*9 = 69 a+9+a*9 = 10a+9 (sub any digit) https://en.wikipedia.org/wiki/Mathematical_coincidence log(1+2+3)=log(1)+log(2)+log(3) follow from 1+2+3=1x2x3 Grégoire Locqueville 2:32 "Maybe one of you can check in the comments" is the new "left as an exercise to the reader" :) Scaling the equations at this time code: https://youtu.be/IguNXoCjBEk?t=256: length, area and "volume" start out the same with radius 1: length=area=volume. When you scale by r, these values scale in this way Length = length * r, Area = area * r^2 and Volume = "volume" r^3. Therefore, Length = length * r = area *r and so (multiply through with r) Length* r = area *r ^2 = Area, etc. Typo spotted: At the very end, in Brahmagupta's Formula the third bracket should be (A+C+D-B) not (A+B+D-B). X minus Y maths t-shirt: Sadly the etsy shop I got this one from seems to have disappeared (Pacific trader). There is what appears to be a ripoff on zazzle by someone who does not know what they are doing :) https://tinyurl.com/24vrzpu9 Nice variation of the t-shirt joke by one of you: M - I - I = V :) The Chrome extension I mentioned in this video is called CheerpJ Applet runner. Music used in this video: Aftershocks by Ardie Son and Zoom out by Muted Enjoy! Burkard