Geometry: Viviani's theorem | Visualization + Proof |
HTML-код
- Опубликовано: 16 сен 2024
- Viviani's theorem basically states that the sum off lengths of 3 lines, drawn at 90 degrees from the sides of an equilateral triangle to any inner point is always equal to the height.
saw this theorem online and thought that I would program a nice and simple visualization for it. What do you think?
Click the link below to interact with the sketch that I programmed:
www.openproces...
_________________________________________________________________
Support me on:
/ think_twice
_________________________________________________________________
Any further questions or ideas:
Email - thinktwiceask@gmail.com
Twitter - / thinktwice2580
_________________________________________________________________
Programs used:
- Processing
- Adobe Premiere Pro
_________________________________________________________________
MUSIC:
• Ether - Silent Partner...
Extremly underrated channel!!!
I agree!
Damn the proof was way simpler than I expected.
I'm sad that what I enjoy in a few minutes must take you so much work. Incredibly fantastic content. If you're enjoying it, please keep going - I'll have to become a patreon.
thank you. Any kind of support is greatly appreciated~
@@ThinkTwiceLtu it is true only for equilateral triangle?
@@sajaltuley1578 I'm no mathematician. But, I'd say most likely, yes. Because, in order for it to work, all sides probably should be the same length and each angle should be the same. This is just a guess, though. Since that would also make each side an equal distance from the center point, which is the initial position of the dot.
In the middle of my binge watch of your channel. Your channel is going to blow up soon. Like 500k by 2019 or something?
Sam Nel haha, well that would be pretty awesome but I think it's quite unlikely. thanks for support
It will blow up. Especially now after 3blue1brown mentioned you in one of his vids.and Your stuff is really good :)
i came her because of 3b1b
think twice is cool!
Even at 500k, this channel would be underrated
Sadly, no
I think there is a better explanation that uses pure visualization, no area formulas. So starting with the three perpendicular lines, first draw a horizontal line through the point. This splits the triangle into two sections: the base and an upper equilateral triangle. Rotate the upper triangle 60 degrees clockwise. Now draw another horizontal line through the point (which has been rotated 60 degrees as well), splitting the upper triangle into another two pieces. Rotate the topmost triangle 60 degrees clockwise. Now we have rearranged pieces of the triangle such that the shape of the triangle hasn't changed, but all perpendicular trisectors line up vertically, showing that they add up to the height
Maths is so fun a beautiful
Nice alternate proof.
I guess when you rotate the upper triangle, you rotate around its center. I think the angle of rotation is actually 120 degrees rather than 60.
I wish I could visualize that too
But the upper triangle is not equilateral.
@@betabeast12 You are wrong. The upper triangle would be equilateral too. Nice proof tho.
this is the only one that I can explain before you tell me how to haha I'm proud lol
Lucy Luo you're too good tbh
thank you!
Lucy Luo just kidding 😁
No, I am also able to.
bamboozled
Your videos explain so well, and I liked the fact the the point wasn't "static", so it was always moving so we could see that it was always true =D.
And again the music fits well in the video! hahaha
Nuno Mateus Hey, glad you liked this one. I tried to make the explanation as simple as possible. The making of this video had many trials haha.
I believe! And at the end you always manage to do it the best way!
Btw is this music from a famous movie or something? it looks like! I'm starting to like it!
tbh I don't know where the music is from. I found it on RUclips by chance haha. But ya I think it sounds nice
Damn! I'd never have thought that it'd be this simple a proof... I mean, just 2 steps???
very lucid and simple... moreover very interesting also
I've always been ass at maths why do I find these videos so interesting
I just love you animations and your explanations are so simple and cool, thanks!
estuardoremi great to hear that!
I love this channel... This channel has both mathematics and science, which I love. And the way of the video is interesting, unlike other channels whose science and math content are boring...
Why isn't this channel not popular
Excellent video, I'll never forget Viviani's Theorem
Beautiful proof. Congrats!
Very nice and simple proof. Video helps understand the proof very easily. Well done
You made this theorem a tablet. I watched it and it took less than 2 mins to understand. The world needs more doctors like you
You deserve MILLIONS of subscribers because you show the world what math is really about.
wait no cause this is actually so cool and just really fascinating I love the visualizations too!1!1!!
I've been following your channel for a while and never failed to be surprised by the elegance of your animations -- presenting mathematics so well visually and simply! Thank you for you amazing talents~
This is just so beautiful.
There's a nice way to do this without areas - notice that you can shrink the triangle on one side until that side touches the point, and this reduces the height by the same amount it reduces the sum. By doing this twice, the point is at a vertex of the triangle, and the one remaining non-zero term in the sum is the length to the opposite side, which of course is the height of the remaining triangle.
learned something new, thanks!
Curiously, I discovered this theorem myself about 30 or 35 years ago, and later learned I had not been original.
However, my original theorem spanned not only triangles but any planar regular polygon: the sum of distances from an inner point to all the sides of a regular polygon is invariant. If a polygon has an even number of sides, it is obvious that the sum of distances from an inner point to both a side and its opposite is constant, hence, so is such a sum of distances to all the sides. But I didn't know of any such easy proof when a regular polygon has an odd number of sides. Now your video and proof give me a way to prove my general case.
Super nice video!! Your efforts are really appreciated, if you could make a video like these for all geometry theorems in Olympiad it would be pretty cool and I guess would blow up pretty fast. I know it takes a tremendous amount of efforts so thanks!!! (PS how do you animated your videos?)
This channel deserves more attention
THIS IS THE COOLEST THING I'VE SEEN IN A WHILE
:)
truly one of the moments of all time
Awesome.
Your video prompted me to an alternate visualization (though it may be more difficult to establish). At 0:57 when black lines replaced coloured perpendiculars, I began to see the animation as an observer hovering over a tetrahedron (perhaps the width of the perpendiculars being less than the width of sides helped create that 3D effect). If one could establish that the collection of triangular projections seen by an observer hovering over the tetrahedron at different angles would cover all possible collections of triangles made by shifting the point in the triangle, one could prove that the sum of perpendiculars is constant.
Subscribed. I don't subscribed to channels that easily. I sometimes unsub from time to time. But when I stumbled upon this channel, insta-sub!
Truly great videos. I've seen every video. Most more than once. Please keep making more!
I could visualise your hardwork too along with this content
Wow, how did I JUST find this channel when looking for some inspiration for a lesson?...
That was so quick and simple that one ends like doubting that this could be the real answer lol
hey think twice ! if you reading this than thankyou for great explanation and efforts
Thanks for watching:)
It is not new to me. But I love how you demontrated the problem and solve it. Hope you keep up and mix in with some more complicated problem.
I feel dumb now for being surprised at how simple that was. Well done!
Really enjoying the telltale music
Beautiful
Love this channel
Excelent theorem, excelent proof, excelente channel! Thank you
when explained in a way that makes sense how, infinity simultaneously is forever and finished, ....all problems will be solved
anyone agree
Gorgeous 💖
Just awesome ❤️
Nice 👏👏
You earned a subscriber.
Another gem. How's your health? Continuing to improve I hope!!
dubarnik thank you. Well it's still pretty much the same as before. But hopefuly it will get better soon.
get better xx
Lucy Luo thanks ☺️
Nice! Very interesting... well done
breno moraes thank you:)
Very informative and beautiful
Area of any triangle is (base × height) ÷ 2.
So break up an equilateral triangle with height "H" ( 🔺️E ), into 3 triangles of different heights ( 🔺️1, 🔺️2, and 🔺️3 ).
We are trying to prove that the sum of the 3 perpendicular distances from any point inside an equilateral triangle must equal the total height of the equilateral triangle, by using the proof:
Area of 🔺️1 + Area of 🔺️2 + Area of 🔺️3 = Area of 🔺️E.
Each perpendicular distance from a point inside the larger equilateral triangle can coincidentally be used as a value of each of the 3 smaller triangles' heights.
The heights for each one is: h1, h2, h3, and H.
And each triangle 🔺️ has the same base, "B".
Each area is:
( base × height ) ÷ 2.
And so it must be true that:
((B×h1)÷2) + ((B×h2)÷2) + ((B×h3)÷2) = (B×H)÷2,
They all are being divided by 2, so it must also be true that:
(B×h1) + (B×h2) + (B×h3) = B×H,
And the base, "B" is the same for all triangles in this case, so factor it out.
Therefore it must also be true that:
h1 + h2 + h3 = H.
So Viviani's Theorem is proven mathematically that the 3 perpendicular distances at any point inside of an equilateral triangle must equal the total height of the equilateral triangle.
This is such a great channel. Subscribed!
Great thanks a lot for this video 👍
I saw complicated in the notation and hypothesis
Wonderful.
with this music all is more epic
why is so underrated?
Beautifull
love it
Really good
Thanks so much
Amazing
Beautiful!
Met your channel by a comment on another video, the you tube algorithm is failing you! I demand viewing justice!
Very satisfying!
Excellent content.Keep up the good work
thank you!
Man your animations are awesome
Awesome
와 진짜 지린다 유익한 영상 감사합니다
Mathematical observation shown here are like lost spells which every Wizard of math wants to find. :)
brilliant
I like how catchy this is in x2.
B-E-A-utiful
That blew my mind.
This is beautiful. :-)
Kylie Estrada ☺️
I dunno, just prove it for the edge cases and then use the mean value theorem??
Love it!
This is great content!
Can you do one with the given point not moving and move the triangle instead? I want to see a three dimensional analysis
This damn song gives me sad nostalgia
More of this please
erozi OK
1:00 that looks a lot like a 3D triangle-based pyramid, wonder if it is related
I proved it by adding the areas
Great doing well job keep it up I am fan of it,sir please upload video on ramanujan numbers
Is it just me or the triangle moves in 3D when the sides are colored? O_o *soooo good*
The sad music made me cry
Thanks makmak krub
Sir viviani theorem is only valid for only equilateral triangle right ..
And thank you for the video sir 😌❤️❤️👊
why does that to be an equilateral triangel only, same thing can be done for scalene right?
So good
Neat.
those are not "any point" inside the triangle are only the points in that circumference inside the triangle.
Question
When we put the point on the bottom of the triangle, surely length point-to-top = height.
Right?
That, plus other lines' length wouldn't the total length be longer than the height then?
if you put the point at the bottom of the triangle, the length of a line from the base of the triangle to the point would be 0. So there would only be two lines to add up, which will be equal to the height of the triangle
Think Twice oohh I get it now . Thanks
At first i read "Vivaldi's Theorem"... I already wondered if he was a mathematitian next to his Musical career...😂
Area Theorem❤❤❤❤❤❤❤❤❤❤❤❤......................................
This was more dramatic than it needed
Awesome!
Mohammed Sharukh :)
Or you could draw a line straight from tip to the bottom to find the height
What if the triangle is not an equilateral triangle
Nice!!!
Hello guys this is your daily dose of the internet