Simple yet 5000 years missed ?

Good news! You really can still discover new beautiful maths without being a PhD mathematician.

Stumbled across this one while working on the magic squares video. Another curious discovery by recreational mathematician Lee Sallows. A simple and beautiful and curious fact about triangles that, it appears, was first discovered only 10 years ago. Really quite amazing that this one got overlooked, considering the millennia old history of triangles.

Wiki page dedicated to Lee Sallows
His personal homepage
The relevant subpage

music: Campagna - Adventure of a Lifetime

Enjoy!

Burkard

[asheep]

Sometimes you wonder how mathematicians come up with things...
Other times, you wonder how mathematicians don't come up with things...

[marksteers]

I love the second "simpler" proof. It is intuitive and I can even explain it to members of the family who are not true maths lovers.

[zzzaphod]

This is a great length of a Mathologer video, nothing wrong with this! Thanks

[JeanYvesBouguet]

The duality relationship between the triangle and its folded form is simply beautiful. As a triangle lover, I absolutely love this video. I cannot believe this was not known.

[rv]

Very nice theorem!
This folding process gives some sort of "sides-medians duality":
-The sides of the folded triangle are each 2/3 the length of the corresponding medians of the original triangle;
-The _medians_ of the folded triangle are each 1/2 the length of the _sides_ of the original triangle.
This proves the 1-time-folded triangle is in general not similar to the original one, but the 2-times-folded one is similar to the original one, with a lengths ratio of (2/3)*(1/2)=1/3.

[davewilson]

One of the best math channels out there. Your glee is contagious!

[WK-]

I did a bit of trigonometry to express the six angles with the coloured dots in terms of the angles of the given triangle. Here's what I figured out. (I'm sure this is known to the triangle experts.)
With the usual notation, let's call A, B, C the points of the triangle, a, b, c the edges and alf, bet, gam the angles. The median through A divides alf into the angles alf_b and alf_c (to the side of the edges b and c respectively). Similarly, bet=bet_c+bet_a and gam=gam_a+gam_b.
With this, one gets:
cot alf_b = 2 cot alf + cot gam,
cot alf_c = 2 cot alf + cot bet
and two similar pairs of equations. (The proof uses the law of sines and the addition formula for cot.) Btw., it can be checked that cot(alf_b+alf_c)=cot(alf).

Now the folded triangle has angles
alf_F = bet_a + gam_a,
bet_F = gam_b + alf_b,
gam_F = alf_c + bet_c,
and one obtains
cot alf_F=(-cot alf + 2 cot bet + 2 cot gam)/3
and two similar expressions for cot bet_F and cot gam_F, i.e. a linear relation between the cotangents of the angles!

So, if one forms a 3-vector from the cotangents of the angles, then the folding operation from the video is the multiplication of this vector with the 3×3-matrix M which has -1/3 on the diagonal and +2/3 in all other entries. This matrix satisfies M^2=1, reflecting the fact that folding twice reproduces the triangle up to size.

[mmathematix]

Another gem from Mathologer. It's because of Mathematicians like you out there, Maths is still beautiful and elegant.

[jonadabtheunsightly]

You had me going at the beginning. Because of the particular choice of original triangle, you briefly had me wondering whether the "folded" triangle might be (geometrically similar to) the mirror image of the original. But no, not in the general case.

[N]

Beautiful theorems. Elegant presentation. Bravo!

[markfacebook]

Seeing the way triangles perpetually fold into themselves is oddly beautiful.

[LeoStaley]

This is my favorite mathologer video in a while. Quite easy to digest, and beautiful.

[moirai]

Thanks! Lovely reminder why I love elegant mathematics like this.

[negsode]

Amazing quality kept for another Mathologer video. Thank you so much for spreading glorious mathematics ideas to mathe-maniacs like us, Mathologer!

[kamaljain]

beautiful! thanks!!

medians of median-triangle give back the scale-down sides -- i also re-discovered this in middle school, when i tried to compute the formula for median lengths using pythagoras and area formula, and noticed that it was a reversible formula, in the sense that you can apply the same formula to get back the sides, if the median lengths are known, with a scaling factor.

now seeing the animation today looks very beautiful.

[maxmn]

What a nice story from the 2D world. Thanks for finding time to share it with us.

[Lightnx]

I like the first proof (the one with the angles) because it also shows immediately with almost no extra work that folding again yields back the original triangle.

[bentationfunkiloglio]

The dot proof is more emotionally satisfying. :)

[nosy-cat]

I think I'd honestly prefer the first proof, but I was too busy shouting at the screen about the second proof to enjoy it.

[bitcoinwallet-tjue]

Loved the dotted one.... really gave a good overview after the flipping.
Thanks for sharing.