Thales's Theorem

hello today we are going to make fun decorative circles and use a festive new way to find their centers join me on Michael Stevens living [Music] is there anyone out there who does not love making circles just kidding I love making circles and today we're going to make some to make our first circle you will need one company sorry this is a family show a comfort a come but contains a sharp point and a marking implement that always maintain the same distance from one another and that is the definition of a circle the set of all points that are the same distance from a given point that given point will be wherever I put the sharp point boom right there now as I rotate the Kumbha around the marking implement will mark all the points that are the same distance from that Center what a wonderful circle but where is this circle Center well it's pretty easy to find out because the come but left a tiny divot in the paper I don't know if you can see that but that allows me to know exactly where its center is now a diameter goes from two points on the circumference through the center because I know where the center is I can use a straightedge to find the circles diameter beautiful this is a fantastic way to make a circle such that it's very easy to divide that circle into two congruent pieces but you don't always have a comfort and a pot isn't always the right way to make the circle that you want so let's make a circle in a different way and see if we can find its center what I'm going to do today is use a circular object from the curiosity box best of addition if you already got box ten which contained the ie in puzzle then you're in luck you already got it but that box is now sold out if you order now you will instead get the best of box and you'll be added to the list of people who will definitely be getting box number eleven if you don't sign up you might not get an eleven because we only make a certain number the best of box is truly full of the best things it's not just like leftovers we couldn't sell we ordered extra of the stuff we really like one of those things is ynx cipher wheel i did a video about how this one works so check that one out but the thing I love about the cipher wheel for this project today is that it is a circle let's use this to make a circle by tracing if I do that I will not have a divot in the center but that might not be a problem I'll show you why let's go ahead and make a circle it's gonna trace around the outside of our cipher wheel oh what a beautiful purple color this is gonna be a really good circle lovely but where's the center I really just kind of have to eyeball it let's see how well I can do it looks like the center I'm gonna guess using this purple marker I'm gonna guess that the center is right there now I'm not really happy with that but let's see how far off I was how do we find the center well one way is to use fales theorem all you need is a right angle for that I'm going to use something else that comes in the best of box the book the elements book ooh beautiful every single element displayed and celebrated the way it should be this book is fantastic because it has a nice sharp right angle to implement fales theorem all we need to do is put our book inside the circle and line up the vertex of its right angle right on the circumference of the circle looks pretty good now we take that vertex and extend to raise along the edge of the book just like that and just like that wonderful now fails theorem tells us that the two other points where these rays contact the circumference are diametrically opposed they are on opposite sides of the circle and a line passing through them will pass through the center let's draw that line I'm going to draw it in orange here it is and as you can see my guess does not land on that line that's unfortunate but you know life is all about mistakes and it's all about being imperfect and bad so good work everyone let's now draw another diameter to find the true center of this circle two different diameters will intersect at the center so let's go ahead and find another diameter I can just put the books point anywhere else I want that looks pretty good let's draw out our rays beautiful and beautiful okay now where did these rays intersect the circumference right there at their vertex and then right here and right here let's now draw another diameter here comes our diameter and there we are where these two diameters intersect is the center of the circle right about there so as you can see I was a little bit off but that's okay let's see how far off I was let's see oh whoops this is my short-sighted ruler is about a centimeter off thank goodness for Thales theorem if you would also like a ruler that doesn't quite use its space optimally WASC is your source my friend Dory makes these and they're pretty fun but here's the question why does fails theorem work why is it that a right-angle inscribed inside a circle contacts the circumference of two other points that define the diameter well I'm glad you asked because that's why this episode is being made there are more neat and rigorous proofs linked down in the description below but I hope in this episode to show how one goes about convincing oneself that fails theorem will always work we need to begin by postulating some ground rules some axioms things that we will declare and accept to be true without proof theorems like Thales theorem are inevitable consequences of the axioms we choose our first ground rule will be defining what kinds of things are isometry x' in universe and isometry is a transformation that preserves distances a transformation is literally just a change for example here's a Pentagon now every side of this Pentagon has a certain length and we could measure that length using a variety of different techniques or tools but now I'm going to transform this Pentagon by translating it and rotating it ooh it's different now but yet every side still has the same length and every internal angle is still of the same measure in fact no matter what I do no matter how I rotate or how I translate this Pentagon its distances and angles don't change now this may sound trivial and obvious but unless it is true fales theorem will not be guaranteed now let's talk about congruence this Pentagon and this Pentagon are congruent congruent literally means that they can fall together if I lay them directly on top of one another they exactly coincide if neither had thickness they would be identical in every way right now composed of the exact same collection and relationships of points equal but congruent is a little bit different because now these two Pentagon's are not equal yes they have the same side lengths and the same internal angles but one Pentagon is over here and one Pentagon is over here they are made of entirely different points so they are not equal instead they're a softer version of equal they're simply congruent two things are congruent if every corresponding measurement of both comes out to identical amounts warning in this video I will occasionally say that two things like two different angles are equal when what I precisely mean is that they are distinct from one another in space but have equal measure that is identical sizes okay so let's now move on from Pentagon's to triangles what do I need to know what is sufficient for me to know to say that two triangles are congruent ruined well if someone tells me that all of their sides are the same length and all their angles are the same well then I know but what if I'm only told the lengths of two sides of the triangle and the angle between those two sides well as it turns out that's enough because look at this how many different ways can I make a triangle using these two side lengths and this angle between them well let's see I can connect both of these lines there's a triangle that's actually the only way I can make a triangle with this side this angle and this side I can translate them and I can rotate them but the final third side will be the same just translated or rotated identically so if two triangles have two sides of the same length and the same angles between them then without knowing anything about the third side I can already conclude that both of those triangles are congruent there are third sides and all angles in them are also the same size now that's a very important property of our space let's talk about some of these consequences I have here two quadrilaterals these are actually some template pieces from when we squared a torus check that video out if you haven't yet these two quadrilaterals are congruent their side lengths and the internal angles are all equal I can translate one of them or rotate one of them and the congruence is still there because these transformations rotation and translation are ossama trees that's exciting but let's start implementing these axioms to get closer to Thales theorem let's take one of these quadrilaterals and trace two of its sides just like that now if I translate this quadrilateral I know that wherever it winds up it will be congruent to its state before let's translate it down like this we've just moved it so that this line this side is right in line with the old side and then this line is parallel to that line see how that works our translation moved every point the same distance from where it used to be and we will declare that that will always be true even if both lines are extended infinitely in both directions so the two lines will never meet they are parallel so interestingly we now have two parallel lines and a transversal let's extend these lines and talk about what we know about their angles beautiful beautiful and it will make this longer perfect now what we know is that there's an angle right here I will call that angle a and that angle is equal to the internal angle of our quadrilateral but remember to create this angle right here all I did is translate the quadrilateral so because translation is an isometry in our space I know that these two angles are the same moving the quadrilateral from here to here did not change that internal angle so these two angles are both a they are corresponding angles meaning that they're both in the same location relative to their vertex is here the intersection points ay and ay are both in the upper-right quadrant of their intersection if corresponding angles are equal that means that this angle which I will call B is equal to its corresponding angle B but now you might wonder what about this angle well here's one thing that we will define the angle measurement of a straight line is 180 degrees if that's true then this angle the one defined by this line right here is equal to 180 degrees and since angles a and B sum up to create this straight line the measure of angle a and the measure of angle B must sum up to 180 degrees but now take a look at this angle right here it's also defining a straight line with angle a so if a plus this mystery angle equals 180 degrees and a plus B equals 180 degrees then the mystery angle must also be equal to B angles that are opposite one another through an intersection point are also equal this means that this angle is a it also means that this angle is a and this angle is B now that we have convinced ourselves of some properties of a line intersecting two parallel lines we are ready to tackle the triangle let's go ahead and draw two parallel lines I'm gonna use this tool for that beautiful here come my two parallel lines that one and that one ooh they're never gonna meet take that lines permanent loneliness now let's draw a line that intersects them it does not matter what angle we draw the line at I'm just gonna find something that I find pleasing and my guests are going to love that one it's fantastic now we know that when two lines intersect angles that are opposite each other through the vertex are equal which means that if this is angle a then this is equal to a it's measure is also a but now let's draw a second line intersecting these two parallel lines to create a triangle I'm gonna use blue for this line I'm gonna make a triangle by having this line pass right through that vertex I can choose any angle I want think I'm gonna choose something nice and scalene or as scalene as possible here comes that line beautiful wonderful now it's actually more that we can say about this Orange Line we know that corresponding angles will be equal and so the upper right quadrant here at this intersection is a which means that the upper right quadrant here is also a a now let's focus back on that blue line we know that the angle here corresponds to the angle up here between the blue and the black lines so if we call this one measure B this one has measure B finally again because vertical angles are equal this angle will have a measure of let's say C which means that this angle opposite it will also be C and finally vertical angles being equal the angle opposite B through the vertex is B now what's very fascinating here is that we've discovered something about a triangle remember that the measure of a straight line is 180 degrees that means that this angle right here is 180 degrees and this angle is the sum of three angles a plus C plus B a plus C plus B equals 180 degrees a plus C B equals 180 degrees the sum of the internal angles in any triangle will sum to 180 degrees we're getting really really close to completely convincing ourselves of fales theorem the final piece we need involves isosceles triangles and isosceles triangle is a triangle that has two sides or under some definitions at least two sides of equal length let's make one I will decide on a length for the two equal sides how about how about 10 centimeters that's feels pretty good 10 centimeters so here's one side and then here's the other does not matter what angle they're at so it's pretty easy to construct two ten centimeter lines we will then connect them I won't do this freehand I'll use a straightedge be nice and beautiful I'll use a different color for this third line there we go this is an isosceles triangle because at least two of its sides this one and this one are equal in length this purple or pink side doesn't even matter but it's different than the other two what I want to know is if I can learn anything about the two angles opposite the equal sides those two angles well I can and here's how I can do it let's bisect this angle up here at the top bisecting means I'm going to cut that angle in half and I'm just gonna eyeball this so it's not gonna be perfect but I will declare by construction that I have cut that angle in half here we go here's our bisector beautiful now what we know by definition is that this angle I'll call this angle a is equal to this angle angle a oh it looks like we might have some congruence going on here remember that if two triangles have the same side length another side that is equal and the angle between those two sides is equal then everything about them is congruent all of their angles all of their sides and have side-angle-side congruence here because look we have two triangles now this one and this one they both have the same side length with that black side they both have the same side length here with this blue one and they also share the same angle between those two sides by using side angle side we can declare that this triangle and this triangle are congruent which means that every measure is equal the side lengths are the same and all the angles are the same so these two angles opposite the same sides b and b are equal angles opposite equal sides in a triangle are equal we are now ready to see why Thales theorem works let's draw out a circle and I'm gonna use a come but for this because I want to know exactly where its diameter is beautiful here's our circle our diameter will be a line that goes through that circle across the circle through the center and contacts circumference in two places there's a diameter a and B the center I will label oh now I'm going to pick a third point anywhere on the circumference I could really pick any point on their circumference I wondered I could even pick B or a I could pick this one I could pick that one I think what I'm gonna do is pick this one that's gonna be nice and pleasing now if I connect this point which I will call C to the center of the circle I have drawn myself a radius so line o C is a radius of the circle and of course the distance from the center of a circle to its circumference is always the radius they're always equal which means that the length of OC is equal to the length of a and OB OC equals o ei equals o b now if we connect a C and B to create two triangles those triangles will be isosceles I'll just freehand this if I connect a and C there's a triangle that has two sides of equal length and if I connect C and B I have in another isosceles triangle this side in this side in this triangle are equal now we know that the angles opposite same length sides are also equal so in our two isosceles triangles we know that this angle which is opposite this side I will call angle a is equal to this angle opposite the same length side so this is also angle a and then in our other isosceles triangle I know that this angle and this angle are equal I will call both of them B now we also know that the sum of all the angles inside a triangle is 180 degrees which means that this large pink triangle we have made contains internal angles that sum up to 180 degrees so 180 degrees is equal to the sum of all the internal angles in this big pink triangle what are those three angles well the first one right here is B make sure you know that that's an angle so B plus this angle which is the sum of B and a and then finally a so a plus a plus B plus B must equal 180 degrees we can simplify this by saying that 180 degrees equals we have two A's and two B's two times the measure of B plus two times the measure of a we can pull a two out of this 180 degrees equals 2 times the sum of angle B and angle a now we can divide 2 out of both sides of this equation 180 divided by 2 is 90 degrees and this divided by 2 just leaves us with angle B plus angle a the sum of a and B is 90 degrees they are complementary they come together to make a right angle the sum of a and B is a right angle two points that are diametrically opposed from each other on a circle will create an inscribed angle of 90 degrees if you use a right angle like we did earlier with the book the two other points intersects the circumference that will be part of a diameter Thales theorem thank you fails thank you theorems and as always thanks for watching [Music]

Channel: D!NG

Views: 1,376,893

Rating: 4.9354382 out of 5

Keywords: vsauce, michael stevens, geometry, thales's theorem, thales theorem, theorem, axioms, math, compass, cirlces, triangles, proof, learn, study, school, angles, parallel, maths, euclid, degrees, ding, d!ng, dingsauce

Id: pJwRsoxe3VE

Length: 23min 50sec (1430 seconds)

Published: Sun Jan 20 2019

Reddit Comments

[u/telemira]

Good video but it only shows one direction.

That is, it shows that given a diameter that has 2 rays, one from each endpoint of the diameter to a common point, then they must form a right angle.

This does not show that if there are two rays at a right angle that they intersect at diametrically opposes points. It follows from what he did, but he did not explicitly shows an if and only if statement; rather, he proves an if/then.