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/Curricul...

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/fil...

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-Bou...

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/FG2012volum...

If you liked the rectangle proof of the sum = product identity you'll probably also like this proof of Pythagoras's theorem:
   • A cute visual proof with a great punc...  
I also mentioned this one earlier in a video on this main channel
   • A cute visual proof with a great punc...  

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/pythagor...

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%2...
A different 3d connection (de Gua's theorem)
https://www.mathpages.com/home/kmath2...

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/questi... (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/Mathema...
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:    • Heron’s formula: What is the hidden m...  
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