I would like to thank everyone for the support that I've been getting. As some of you may know my health condition isn't that great at the moment so all the nice comments really help to cheer me up through out the day. Hope you enjoy watching the video and have a nice day :)
Sometimes the beauty of mathematics is hidden and arithmetic proofs don't have the same beauty to it... The content of this channel brings out mathematics in its raw form and discloses the deep meaning hidden in plain sight.. Thank you @Think Twice for all the efforts..
My only problem: Why is it that the squares A^2 + D^2 and B^2 + C^2 (both visually equal) are perpendicular when rotated along the circle. How is this and also, how does this translate mathematically and visually when the squares are not equal? Love the video!
Sy Fontenot for the purpose of the proof, when sqrt(A^2 + B^2) and sqrt(C^2 + D^2) are perpendicular, their hypotenuse equals the circles diameter. I wish he explained that as well
Sy Fontenot because the cross divides the circumference in 4 arcs, in such a way that two opposite arcs will add up to half the circumference. I'll think about a way to prove why this is true and get back to this comment section
@Sy Fontenot: Segment AC and BD are perpendicular, by definition. Now bring their intersection point (i.e. the point inside the circle where A, B, C and D meet) to the edge of the circle, so that ultimately the intersection-point is *on* the circle (and, in the animation, C = D = 0). Now it's obvious that A and B will still be perpendicular, and so the squares will be. Final step: since A and B are perpendicular, the hypothenuse will now be a diameter of the circle according to Thales' theorem. @Think Twice: beautiful visual proof, well done. Nice one to do in class: present the theorem, ask for a proof, let students sweat for 15 minutes, and then show this animation :-).
I still don't understand where we show A²+B²+C²+D² is constant. The video only shows that for C=0 and D=0 the sum is equal to the diameter squared, but not that this is true for any point in the circle. What am I missing?
Patrick Wienhöft: Start at 1:11 where you see two squares a^2+d^2 and b^2+c^2 for /random/ a, b, c, d. Then from 1:14-1:17 it is shown that you can rotate those squares so that they end up as perpendicular squares, i.e. squares on the short sides of a rectangular triangle whose hypothenuse is the circle's diameter. This indeed is where c=d=0 but what you miss is that it is derived from a situation where c and d are *not* 0. So no, the video does not show the property only for c=d=0, as you suggest, but for any combination of a, b, c, d.
When he moves those two lines such that they are at a right angle, then according to Thales's theorem the line joining the other free points are diametric ends.
Its not obvious that they indeed make a right angle once the two chords are moved so that their endpoints coincide, but it does follow from the fact that the original two lines (the lines "AC" and "BD") were perpendicular.
It follows from the fact that the angle formed by two chords is the average of the intercepted arcs. Therefore, the sum of those intercepted arcs must be 180, since the average is 90. Then, when you move one part of the arc and attach it to the other side, the sum is still 180 (half of a circle). mathbitsnotebook.com/Geometry/Circles/CRAngles.html
Because the four points cross to form a right angle at the beginning, which means each pair of opposite segments contains half of the circumference. When you bring the two line segments together, the hypotenuse is then the diameter. By definition, the angle formed at the circumference would be a right angle
@@anshulagrawal633 Not sure if you still have this doubt, but I'll give it a go. You can think it this way: The angle formed by the segments before the transformation is 90º. By the angle beetween two chords formula, 90º = (arcBC + arcAD)/2, so arcBC + arcAD = 180º. After transformation, that relation is still valid. Let us call the angle beetween the "AD" segment and the hypothenuse A, and the angle beetween the "BC" segment and the hypothenuse B. The angle A "sees"(I'm brazilian, not sure what the correct term would be) the arc "BC", so A = arcBC/2 By the same reasoning, B = arcAD/2. so A + B = arcBC/2 + arcAD/2 = (arcBC + arcAD)/2 = 90º So the triangle is a right triangle. The "containing half of the circumference" part would be refering to the fact that the arcs add up to 180º
This is such an amazing site to learn, once again, the concepts we knew only as written proofs. Please keep up the good work and you may want to take a look at the geometrical shapes of the ancient Astronomical Clock "Jantar Mantar"
This one does just assume (even though true) that the two unequal squares finally forms a rectangle at the joining point. But nice video nevertheless :-)
I don't think I'm looking at this right, but isn't that Pythagorean's theorem being used at 1:24? If so, shouldn't the equation be "(A^2)+B^2=C^2"? Pluging (A^2)+D^2 from the bottom square for A and (B^2)+C^2 from the square from the right with C, do you not get ((A^2)+D^2)^2+((B^2)+C^2)^2=Diameter^2? I don't get this...
Jo Wick, Notice how the (A^2)+D^2 is already the square. Same with the other square. Since they’re already the squares, there is no need to square it again to fit into the Pythagorean Theorem. Just substitute them for a^2 and b^2 in a^2 + b^2 = c^2. The diameter is c therefore diameter^2 = c^2. And A^2 + B^2 + C^2 + D^2 = Diameter^2.
Wonderful videos which you are creating, Think Twice! Thanks a lot for this. In this one unfortunally I still don't see why the two segments form a right angle when the endpoints are brought together.
1:16 I think you just showed how A²+B² = Diameter²(Pythagoras theorem). Because you considered only a particular case when C and D are zero. Pls correct me if I am wrong.
For the diameter at 1:21 wouldn't the sides be (A^2 + D^2)^2 and same for (B^2 + C^2)^2 as the sides to find diameter/hypotenuse squared? Which gives a different answer
For this video I used a program called processing, you can make programmable animations using that program. Also I usually use another program called cinema 4D. If you have any other questions feel free to ask:)
I know a person whose actively involved in Processing, he also creates Processing tutorials. His name is Daniel Shiffman, his channel is Coding Train. Maybe you want to take a look. :)
Yes I started working with processing about 2 months ago and pretty much everything I know about it I learned from that channel. He is good at explaining and has a lot of videos
@suyash The point cannot be "anywhere" in the circle - if you look at the very beginning of the video when there is just the circle and on vertical and one horizontal line - these two line have a single intersection point - the loci of which traces a very specific path. My question is what is the shape of the loci this intersection traces out?
it took me a minute to pause the video and do some angle chasing to convince myself it's correct, just take a look at the angles that have arcs in common, and since those angles add up to 90 degrees... you know what follows
I would like to thank everyone for the support that I've been getting. As some of you may know my health condition isn't that great at the moment so all the nice comments really help to cheer me up through out the day. Hope you enjoy watching the video and have a nice day :)
Hope u'll get well and awesome video!
Thank you!!!
超级喜欢你的每个视频，轻松又有趣，尤其是这个！加油，也希望你尽快恢复健康！
谢谢
I had no idea about your health, I hope you recover fast and keep making such amazing videos. :)
Nothing can be more beautiful than this. Man, your videos are what I call as "Happiness".
Thanks for kind words :) I'm glad you like it
Very, very pretty :)
Mathologer thank you!
it's kinda cool that you watch his channel!
I think it would be better if there was the demonstration of the second fum of squares, which can be done more or less easily
I love this kind of little geometry proof. Simple and straightforward and show how beautiful maths is
yes sometimes the most simple proofs can be one of the most beautiful
I discovered this channel yesterday ... the most beautiful thing that happened to me all year.
This is one of those moments when you go like "Awww!" because of the sheer beauty of it. Thanks for making my day!
Ulkar Aghayeva I'm glad you liked it! comments like that make my day haha
Cannot comprehend how nice this is
I'm so glad I found your channel.
This is what mathematics is.
Love the Euclidian geometry proofs, very intuitive and can always be useful somehow hahahha.
Keep going!
Haha ya true. Thanks!
Honestly these videos are just so beautiful
Hence, it gets proven, maths is fun! And, the people who make it fun are great!
Sometimes the beauty of mathematics is hidden and arithmetic proofs don't have the same beauty to it... The content of this channel brings out mathematics in its raw form and discloses the deep meaning hidden in plain sight.. Thank you @Think Twice for all the efforts..
Thanks for watching:)
So beautiful, so elegant
My only problem: Why is it that the squares A^2 + D^2 and B^2 + C^2 (both visually equal) are perpendicular when rotated along the circle. How is this and also, how does this translate mathematically and visually when the squares are not equal?
Love the video!
Sy Fontenot for the purpose of the proof, when sqrt(A^2 + B^2) and sqrt(C^2 + D^2) are perpendicular, their hypotenuse equals the circles diameter. I wish he explained that as well
Sy Fontenot because the cross divides the circumference in 4 arcs, in such a way that two opposite arcs will add up to half the circumference. I'll think about a way to prove why this is true and get back to this comment section
@Sy Fontenot: Segment AC and BD are perpendicular, by definition. Now bring their intersection point (i.e. the point inside the circle where A, B, C and D meet) to the edge of the circle, so that ultimately the intersection-point is *on* the circle (and, in the animation, C = D = 0). Now it's obvious that A and B will still be perpendicular, and so the squares will be.
Final step: since A and B are perpendicular, the hypothenuse will now be a diameter of the circle according to Thales' theorem.
@Think Twice: beautiful visual proof, well done. Nice one to do in class: present the theorem, ask for a proof, let students sweat for 15 minutes, and then show this animation :-).
I still don't understand where we show A²+B²+C²+D² is constant.
The video only shows that for C=0 and D=0 the sum is equal to the diameter squared, but not that this is true for any point in the circle. What am I missing?
Patrick Wienhöft: Start at 1:11 where you see two squares a^2+d^2 and b^2+c^2 for /random/ a, b, c, d. Then from 1:14-1:17 it is shown that you can rotate those squares so that they end up as perpendicular squares, i.e. squares on the short sides of a rectangular triangle whose hypothenuse is the circle's diameter. This indeed is where c=d=0 but what you miss is that it is derived from a situation where c and d are *not* 0. So no, the video does not show the property only for c=d=0, as you suggest, but for any combination of a, b, c, d.
I must be missing something because it is not obvious to me that the line you create at 1:16 is actually the diameter
en.wikipedia.org/wiki/Thales%27s_theorem
When he moves those two lines such that they are at a right angle, then according to Thales's theorem the line joining the other free points are diametric ends.
Its not obvious that they indeed make a right angle once the two chords are moved so that their endpoints coincide, but it does follow from the fact that the original two lines (the lines "AC" and "BD") were perpendicular.
terdragontra do you have a proof for that?
It follows from the fact that the angle formed by two chords is the average of the intercepted arcs. Therefore, the sum of those intercepted arcs must be 180, since the average is 90. Then, when you move one part of the arc and attach it to the other side, the sum is still 180 (half of a circle). mathbitsnotebook.com/Geometry/Circles/CRAngles.html
May I know how could you assure that the two line segments must touch to create a right angle? Couldn't it be acute/obtuse?
You'd stop having squares...
Because the four points cross to form a right angle at the beginning, which means each pair of opposite segments contains half of the circumference. When you bring the two line segments together, the hypotenuse is then the diameter. By definition, the angle formed at the circumference would be a right angle
@@MrNicePotato how each pair of opposite segments covers half the circumference?
@@anshulagrawal633 Not sure if you still have this doubt, but I'll give it a go. You can think it this way: The angle formed by the segments before the transformation is 90º. By the angle beetween two chords formula, 90º = (arcBC + arcAD)/2, so arcBC + arcAD = 180º.
After transformation, that relation is still valid. Let us call the angle beetween the "AD" segment and the hypothenuse A, and the angle beetween the "BC" segment and the hypothenuse B.
The angle A "sees"(I'm brazilian, not sure what the correct term would be) the arc "BC", so A = arcBC/2
By the same reasoning, B = arcAD/2.
so A + B = arcBC/2 + arcAD/2 = (arcBC + arcAD)/2 = 90º
So the triangle is a right triangle.
The "containing half of the circumference" part would be refering to the fact that the arcs add up to 180º
This is such an amazing site to learn, once again, the concepts we knew only as written proofs. Please keep up the good work and you may want to take a look at the geometrical shapes of the ancient Astronomical Clock "Jantar Mantar"
You are the best man, keep it up
SVP thaks a lot!
Impressive as usual!
These videos made me like math again, tyvm
That was lovely !
Pause at 0:14 for smash ball
Lego Luigi eyyyy look at waluigi
euclid confirmed smash dlc
Very nice video sir please upload more videos like this
this would look really cool as a loading screen logo!
Excellent ⬜️🟨🟧🟫⚫️⚪️⚫️⚪️⚫️⚪️⚫️⚪️🚀🌈☮️💟🗽🤯🥂🎬
beautiful!
This one does just assume (even though true) that the two unequal squares finally forms a rectangle at the joining point. But nice video nevertheless :-)
i wish teachers would use this alongside the proofs where they just derive from equations;;;; my maths life would have been soooo much more clearer
I don't think I'm looking at this right, but isn't that Pythagorean's theorem being used at 1:24? If so, shouldn't the equation be
"(A^2)+B^2=C^2"? Pluging (A^2)+D^2 from the bottom square for A and (B^2)+C^2 from the square from the right with C, do you not get ((A^2)+D^2)^2+((B^2)+C^2)^2=Diameter^2? I don't get this...
Jo Wick, Notice how the (A^2)+D^2 is already the square. Same with the other square. Since they’re already the squares, there is no need to square it again to fit into the Pythagorean Theorem. Just substitute them for a^2 and b^2 in a^2 + b^2 = c^2. The diameter is c therefore diameter^2 = c^2. And A^2 + B^2 + C^2 + D^2 = Diameter^2.
Ohhh, okay, now I get it. Thxs!
Oooh that makes way more sense
Wonderful videos which you are creating, Think Twice! Thanks a lot for this.
In this one unfortunally I still don't see why the two segments form a right angle when the endpoints are brought together.
very nice proof
*_[: Yay Smash bros logo!_*
Mind blown!!
So satisfying and nice music
How can we prove that A²+B²+C²+D² is always equal to the diameter² when C and D are different from 0? (1:22)
I was thinking the same. He just showed A²+B²= Diameter² by considering only a particular case when C and D are 0.
But how do you know that works for all values of A, B, C, and D instead of just that particular configuration?
1:16 I think you just showed how A²+B² = Diameter²(Pythagoras theorem). Because you considered only a particular case when C and D are zero.
Pls correct me if I am wrong.
Smile each time a smash ball forms.
Excellent video
thank you!
What is the shape defined by the point at the intersection of the four squares as it moves around inside the circle?
Brilliant
For the diameter at 1:21 wouldn't the sides be (A^2 + D^2)^2 and same for (B^2 + C^2)^2 as the sides to find diameter/hypotenuse squared? Which gives a different answer
So this is my newest source of dopamine.
This makes me wanna cry off joy
Me:I am now an intellectual person.I must share this useful information with my near and dear ones.
My mind: *spinny shapes...whoooaaahh*
It's so beautiful and interesting!
I wanted to ask you, how is this animation made, i.e. what softwere did you use?
Apple Productions for this one I used software called processing :)
Think Twice What frameRate did you use?
Apple Productions 60fps
How do you make such beautiful animations? What software do you use?
For this video I used a program called processing, you can make programmable animations using that program. Also I usually use another program called cinema 4D. If you have any other questions feel free to ask:)
okay, check out this link processing.org you can download it there and also look up different tutorials.
I know a person whose actively involved in Processing, he also creates Processing tutorials. His name is Daniel Shiffman, his channel is Coding Train. Maybe you want to take a look. :)
Yes I started working with processing about 2 months ago and pretty much everything I know about it I learned from that channel. He is good at explaining and has a lot of videos
Think Twice, I'm sure he will be really proud of you. :)
0:14 Super Smash Bros.
@suyash
The point cannot be "anywhere" in the circle - if you look at the very beginning of the video when there is just the circle and on vertical and one horizontal line - these two line have a single intersection point - the loci of which traces a very specific path. My question is what is the shape of the loci this intersection traces out?
That depends on the loci of the two lines
it traces out the entire surface of a square point could even be outside the circle and inside that square
0:14 EVERYONE IS HERE
0:14 Super Smash Bros logo.
0:14 SMASH
0:15 Smash logo
1:15 But, how do we proof that two segments are from a rectangular triangle?
All I see is a smash ball
I keep thinking about the super smash bros logo
What is that creepy music behind the math?
Intressting
You are great
Thank you :)
Think Twice u deserve it.
Try watch this on high, beautiful
balloons pumping in echother
it took me a minute to pause the video and do some angle chasing to convince myself it's correct, just take a look at the angles that have arcs in common, and since those angles add up to 90 degrees... you know what follows
Is there a name for this theorem? I really want to find a formal proof.
hmm sorry I have no idea, let me know if you find out:)
Think Twice okay, thank you! :)
It's proposition 11 from Archimedes Book of Lemmas
IT CANT BE PROOF IF I HAVE NO CLUE WHATS HAPPNING
0:12 smash bros. logo
This coloring reminds me of someone... Who is he?
:'/
This video got great visual for sure. But the proof is not complete :/. At 1:18, why the cord you trace is a diameter ? The video is quiet misleading.
in general l love your proofs, but this one is far from a proof. two important steps are simply supposed. Mind the quality of your channel ;-)
First
best