Viviani's Theorem (visual proof via rotation)
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- Опубликовано: 7 июл 2022
- This is a short, animated visual proof of Viviani's theorem, which states that the sum of the distances from any interior point to the sides of an equilateral triangle is equal to the length of the triangle's altitude. #math #geometry #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #polygons #triangle #mathshorts #mathvideo
This animation is based on a visual proof by Ken-ichiroh Kawasaki from the June 2005 issue of Mathematics Magazine (www.jstor.org/stable/30044158) page 213.
Here is a cool interactive GeoGebra applet to play with this theorem yourself:
www.geogebra.org/m/Jgv9N6gR#:....
To learn more about animating with manim, check out:
manim.community
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Music in this video:
Dark Side Of Our Past by MaxKoMusic | maxkomusic.com/
Music promoted by www.free-stock-music.com
Creative Commons Attribution-ShareAlike 3.0 Unported
creativecommons.org/licenses/...
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Wow, what a great visual proof! Well done.
Thanks. This was a fun one to figure out in manim :)
@@MathVisualProofs I can imagine the Manim challenge :)
Wonderful, as usual. I am currently preparing an exam to become Mathematics teacher in high schools and I find your videos incredibly educational... Thank you for your work
Glad you like them. Thanks for watching!
wonderful explanation of this theorem
thanks
I once discovered a proof of the fact that the centroid divides the median in the ratio 2:3 using geometric series. I used the fact that the centroid of a triangle is the same as the centroid of it's medial triangle. So if we keep constructing medial triangles recursively, the centroid would be sandwiched between the base and opposite vertex of the medial triangle. Both the base and opposite vertex would approach a certain height (equal to the height of the centroid by the squeeze theorem) as the number of medial triangles approaches infinity. I think that would make for a good visual proof.
I'll see what I can do :)
@@MathVisualProofs don't just see, but try it 😊 as seeing is our job! 😉
Bro isn't the ratio 2:1 also did u come up with this by geometric series
Are you considering entering a video into 3blue1brown's Summer of Math Exposition 2?
Yes. I entered one last year (not too long after I started this channel last year). I’ve been working on one again this year. It’s almost done but ended up way longer than my usual videos.
@@MathVisualProofs Thats great to hear. I'll look out for it
Very nice proof! Binging all your videos
@@juancristi376 thanks! Hope you enjoy them all. Been working hard to improve over the last 1.5 year. :)
The best proofs don't use words!
😀
Very nice proof!
Thanks!
Damn 2k views? I got here way before the algorithm makes it explode. Great video
Hah! Thanks. Never sure how the algorithm works. Glad to see this one getting at least some traction. Thanks for checking it out :)
I just came across a channel called Mind Your Decisions.. with almost 3M followers. And it reminded so much of this channel. In fact I think this channel is better! You got the goods of something great - just gotta keep running the marathon!
Thanks as always for the continued encouragement! We’ll see how long I can keep it up. It has been a fun side project for the past year+ … :)
@@MathVisualProofs I know the feeling. I’ve been cranking away since April 2020. Takes time! Slow and steady :)
@@Mutual_Information you came out the gate quick it seems. Very nice video editing and merging of media. I basically only know how to animate :)
Really nice!
Thanks!
amazing
:)
So elegant and convincing.
😀👍
A beauty, you've outdone yourself on this one. Wonderful.
Thanks!
this video was amazing, i understood everything with great precision. keep it up
Thanks! I appreciate the comment. And thanks for watching! :)
Here is another proof, based on a method my friend came up with:
Notice that the length of every distance from the point to the side is a linear function with regards to the point, so their sum is as well.
This sum is equal to what we want in the corners of the triangle, and therefore always equal to what we want.
"Proof from linearity"
You can also have functions in the second degree, which have to be checked in six places. There is a theorem with sums areas in an equilateral triangle, and you only need to check it in the corners and the midpoints of the sides, which all follow from symmetry.
He came up with this method when I asked him for a proof of
[SAB][SCD]+[SAC][SDB]+[SAD][SBC]=0,
apperently known as Ptolemy-Stewart's area theorem, but it doesn't even have a Wikipedia page. I learned that was the name of the identity from ChatGPT.
great
Thanks!
If you denote by d1,d2,d3 the distances to the sides , then one has d1*s+ d2*s+d3*s =2* Area =1/2*s ^2*√3 ,s= sidelength.Hence ,d1+d2+d3=1/2*s*√3= height.This is also a very simple proof.
Beautiful. Alternatively,at 1:10 you can reflect horizontally the segment inside the blue triangle and translate the small segment upwards to align it with the reflected one.
Yes! I find that one takes more thought to believe it fits, but it’s a nice version. Thanks!
Hey! Thats my ancestor's theorem! Directly from my mother's side of the family!
👍
🔥🔥🔥🔥🔥
😀
“Without words”
0:28
Well it's still best to state the theorem with words :)
muito bom!
Thanks!
Maravilha
Beautifully done! :)
Thanks for checking it out!
I feel incredibly stupid. I tried proving this theorem on my own and I did it in a super complicated way using analytic geometry. I see this proof and the other which relies on the sum of the areas and I feel like absolute shit. I've been away from maths for a bit, so that might be a factor
All proof techniques and approaches (including yours) are worthwhile as they move us along to understanding. Don’t let it make you feel bad. Just keep at it.
@@MathVisualProofs thanks! it truly is a noble cause
Keep the point in the center and reposition the two top lines to point down. They both become longer but the bottom line remains the same. So the proof can’t be right.
I’m not sure I understand. Can you elaborate ?
I think the angles of the lines have to stay at 120°. The theorem must have certain set points you have to follow.
The equilateral triangle probably has to be normal and not concave or convex as another set point.
@@germancarranza236 yes. The lines are the shortest lines from the interior point to the edges. They represent distance from point to edge. Thanks for helping out! :)
@@germancarranza236 makes sense
I need to learn math but it kind of sucks
It’s truly a phenomenal subject. I know it gets a bad rap sometimes but it is just too amazing to not love. :)
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