Find The Center Of A Circle With Thales’ Theorem

One of my favorite methods for finding the center of a circle is Thales’ Theorem. This method can be accomplished over long distances with the aid of a three access laser level and string line.

[@m2d5]

A right angle inscribed in a circle will have a diameter as its hypotenuse. Two diameters will intersect at the center.

[@davidsorensen4400]

Bingo

[@larsbkurin1740]

Stonemasons who built cathedrals and fortresses in the Middle Ages began by establishing the unit 1 by driving two studs into the rock or into a flat stone. A firm, level floor was then built around the site. Ruler, caliper, string, chain (rigid), spirit level was all they needed. Their knowledge of how to make circles, ellipses, 90 45 degrees from a line was their magic and secret. In a few places, the studs are saved for repair work. Saw a French documentary about this, if I find it again I will link it. I apologize for the language error, this is written with google translate.

[@benjaminvalenzuela3948]

I would be interested in seeing that documentary.

[@grumpyparsnip]

Cool. I think this would be great to show to my geometry class. One pedantic note. I believe you are using the converse of Thales Theorem: if you have an inscribed right triangle in a circle, then the side opposite the right angle is a diameter. (In other words, Thales' theorem says "diameter implies right triangle" while you need "right triangle implies diameter.")

[@KenFullman]

An important point you kinda hinted at is that the triangle doesn't need to be isosclese. ANY 90 degree triangle will produce a diameter to the circle. I think this point was sort of smudged by referring to isosceles triangles in your explanation of Thales theorem.

[@txtradesman527]

You are correct, I should’ve been more clear on that point.

[@jeffgustafson2131]

It might have been a bit vague on that point, but it is very relevant! Producing that third radius that divides the first triangle in two ends up making two not necessarily equal isosceles triangles (each being composed of two radii plus another side). Great video! I love seeing this practical geometry being taught out in the wild! (I work at a university and am always seeking ways to find motivation and inspiration outside the classroom for our students)

[@billmurray4895]

You can do the same with a 3-4-5 triangle.

[@belomolnar2128]

Mr. THALES from Miletus was the first Philosopher ever. I could not imagine who told him this. He was the first ever. ❤

[@peterhall6656]

As an applied mathematician I sometimes slum it by watching this type of video. Machinists have all sorts of hacks they use but could never prove in any rigorous way but they do work and are based on proper Euclidean geometry. A carpenter once asked me about a rule of thumb he used fo arches and I derived it from basic ellipse properties and it made sense. 17th century mathematics was highly geometric and Newton's Principia is almost unintelligible to modern readers. Indeed, Nobel Prize winning physicist Richard Feynman once tried to replicate Newton's highly geometric proof of his inverse square law of gravitation and it defeated him because Newton relied on obscure geometric properties that we simply don't learn these days because of analytic geometry etc. Newton did a geometric derivation of the shape of minimum resistance in a fluid and althougn it was obscure he got the same answer a fluid dynamicist would get woth modern techniques. Geometry is really powerful and only involves simple tools.

[@gsbedfordshire591]

Brilliantly explained and to the point. Excellent! Thanks.

[@txtradesman527]

Glad it was helpful!

[@fullgasinneutral9368]

Here is another tip - how to create a right triangle when you have no rulers or levels or any equipment - only a long string and a knife. Use anything as a yardstick, and arm, a leg, a branch. Heck, stick two pegs in the ground, or use two rocks and define the distance as one unit length. Cut the string into three cords: 3, 4 and 5 units of length. Tie the ends of the cords together (3 to the 4, 4 to the 5 and 5 to the 3) and when you pull on the knots to the maximal extent, you got yourself a right triangle with the hypotenuse is the 5-units length cord. Of course, any multiple of the Pythagorean triplet would work: (6,8,10) , (9,12,15) etc. Pythagoras was a complete cultist loon, but a smart guy nonetheless.

[@wellscampbell9858]

Here's a handy add-on. To quickly get your cord lengths, take your string and anchor it on the corners of the 3-4-5 triangle you just layed out. Pull tight and mark at the corners. Cut at the marks and knot the ends. Now you're ready to lay out your triangle.

[@anthonycannet1305]

Thale’s theorem is just the inscribed angle theorem in reverse. The right angle used is an inscribed angle, and the inscribed angle theorem states that the angle AVB is equal to half the angle AOB, where O is the origin, and A, V, and B are points that lie on the circle. So given three points on the circle and the angle 2ø between two radii, you know the inscribed angle at the third point is equal to ø. But you can also go in reverse. Given an angle of 90° inscribed in the circle, the other two points form an angle of 180° with the center, so drawing a straight line between them must intersect the center. Then you just do that twice to find the intersection of the two diameters.

[@spelunkerd]

Yes. The most stunning practical demonstration of this is to draw a line AOB through the center of a circle. Then put a pencil on a third point, V, anywhere else on the circle. Even as you move that point V around the circle, the angle AVB remains exactly 90 degrees, always exactly half of AOB.

[@simpleman283]

I like the video. I'll save it just like I did the other one. I already knew about this theorem somewhat, but this explanation was really good. As for that guy down there beating his drum about where the center is. Well, people like that are better off ignored. Any one watching this video can see the good value here.

[@simpleman283]

Oh yea, I'm definitely part of the Blue Collar Brotherhood. 20 + years doing carpentry.

[@txtradesman527]

I appreciate that, and I think you’re most certainly right regarding my new friend. Thank you for taking the time to watch.

[@tim71291]

Thank you for the very direct route to explaining this! The paint story is so believeable in that I heard my father talking about how his days at nuclear sites- everything was so technically right yet painfully expensive when two surfaces did not meet because of such things.

[@JobBouwman]

At 0:56 you could have used the endpoint of one of the existing two lines, to draw the third in a right angle, of which that third is parallel to the first. The diagram becomes simpler and easier to understand.

[@edwardblair4096]

The other way I have heard this is that if you start with a semi-circle and draw a line from one end to any other point on the semi-circle and then from that point to the other end of the semi-circle the angle between those lines will allways be a right angle. You are using this property in reverse by putting the vertex of a known right angle on a circle. The intersection of the right angle with the circle will be a semi-circle which lets you draw a diameter line, which by definition goes through the center of the circle. Do this a second time to prodoce a sevond diameter line and the intersection of the two diameters is the center of the circle.

[@billmurray4895]

Great example of basic geometry!

[@andrewzepf7510]

It works because of another theorem whereby the if an angle has its vertex on the circle the the angular measure of the arc enclosed within the rays of the angle will be twice the angular measure of the angle itself. Using a square (90 degrees) therefore encloses an arc with an angular measure or 180 degrees.

[@saranevillerogueart9627]

Make a square fit your circle into it. Or draw your circle. Draw the square around it. Make an X across them

[@txtradesman527]

How exactly is that easier? I’m genuinely asking.

[@RuudAlthuizen]

Making an actual square can be quite challenging, meaning all legs are the same length, and all corners are 90°. You’d also need to measure the diameter of the circle, for which you’d also need to know the middle of the circle to do that measurement accurately.

[@saranevillerogueart9627]

It easier for me . Than all those complicated mathematical like angles. I jus draw circle. Make a straight line right next to it. Then use metal triangle with a 90° angle in it. To draw A connecting 90° angle to it right up to another edge of the circle the make 2 more on the other places on the circle to connect the square then draw an X from the opposite corners of your square. Easier to do than all this other stuff and easier to remember

[@txtradesman527]

There are no complicated angles to remember.

[@robertstuckey6407]

With a carpenters square i can do the method in the video using 6 marks (dont need to draw the legs of the triangle and you only need a minimum of 4 distinct points)

[@michaelmartin5995]

Nice demonstration. I must deploy mathematics constantly in my daily work. (Customers seem to like things built well.) :)

[@rossk4864]

Very interesting discussion of geometric principles. In your story: considering that the unfortunate structural engineer of record on this project actually thought that builders could construct the roof to within 4-mils of design elevation is both quite amusing and quite distressing at the same time! To construct it to within 1/2" of design would be impressive.

[@cafemolido5459]

Draw two segments, segment ends touching the edge of the circle, shoot a 90 degree line from center of each segment, they'll cross at center of circle

[@JeffThePoustman]

All this also comes from the Lord Almighty, wonderful in counsel and magnificent in wisdom. -Isaiah 28:29

[@TERRYBIGGENDEN]

Great story and totlaly believable Different attitudes in different professions. :-)

[@syedghulamhaidershah582]

Black pencil on green paper makes poor vision.

[@txtradesman527]

It never ceases to amaze me the things that people will complain about.

[@jwm6314]

Set resolution to max. The compression makes it fuzzy, max resolution is clear.

[@swedishpsychopath8795]

Why not just draw a rectangle or square inside the circle and draw lines from the corners? Where they cross is the center of the circle.

[@christianaxel9719]

I think is easier and only needs compas and ruler is to trace two different chords, then trace the perpendicular line at the middle to each chord using compass and ruler (open compass slightle more than half teh chord then trace two arcs pivoting one from each point in the circle in the chord, then unite the two crossing points with ruler and extend it as necessary into the inside of the circle), then the point where they cross is the center of the circle.

[@RuudAlthuizen]

I think that’s also based on Thales theorem, just using a different tool.

[@chrisgee5893]

Or just use the set square to box the circle and draw the diagonals; where they cross is the centre.

[@davido3026]

Draw any inscribed triangle in a circumference, then Trace the 3 mediatrixes. The point where they meet is the center of the circle! The prthocenter!!!

[@txtradesman527]

I think you mean orthocenter. And I have no idea what a mediatrix is.

[@romailto9299]

Very useful! Now are there 2 thales theorems?

[@txtradesman527]

Yes, the Proportionality Theorem.

[@txtradesman527]

There’s a few more theorems attributed to Thales as well, including the theorem that proves the base angles of an isosceles triangle are equal.

[@sidster64]

You just be a Millwright LOL great video

[@adrianwilliams763]

Love the story.

[@ningayeti]

Watching on my phone the pencil marks on the green paper are invisible. Might be different on a computer.

[@peterbaruxis2511]

Did I miss it or was the laser level the way he found the center of the 60' dia silo? was this to lay out a future silo on a pad bigger than 60' giving him a place to physically put the instrument- i don't see where he could have put the instrument inside of an existing silo- or outside of an existing silo (where it would have been useless anyway.)

[@txtradesman527]

That’s a good catch. What I had to do was use a 6” spacer to mark out a circle on the concrete that decreased the radius by 6”. Then I was able to set my laser on the smaller circle to pick up two diameters.

[@peterbaruxis2511]

@@txtradesman527 Cool.

[@Bob94390]

The proof you gave for Thales' theorem only concerns the very special case of an isosceles triangle. Thales' theorem is much more general.

[@txtradesman527]

I’m aware.

[@robinbrowne5419]

That's pretty cool 👍

[@Traderjoe]

I would like to see if there’s a way that if you had like an empty oil drum and you wanted to find the center at the bottom of it, how the center could be determined

[@thomasw.eggers4303]

Turn the drum over and make your marks on the bottom, exactly as shown in the video.

[@economicist2011]

The kind of set square he's using is particularly well-suited for inscribing right angles on the inside of a bounded circle, but you're right that you won't simultaneously reach both points on the diameter unless you have a set square of exactly the right dimensions. You'd have to mark your inscribed right angle first, and then construct the proper extensions on each side to reach both points on the diameter. You can do that with another, longer straightedge. If you're uncomfortable with the potential for imprecision when continuing a short line segment with a straightedge over a long distance, you could alternatively draw any two wide chords across the bottom of the drum, and then construct perpendicular bisectors of each. They'll both be diameters, and will thus meet in the center.

[@saranevillerogueart9627]

Draw a square around the circle. Use two straight sticks from corner to corner across top of the barrel

[@thomasw.eggers4303]

@@saranevillerogueart9627 Your geometry is correct, but the method in the video is easier. (I suggest that you actually try it both ways.) Constructing your square makes your suggested method harder.

[@TimeSurfer206]

As thomasw said, flip it over. And go ahead and use your square on the inside of the rim. Neither the square or the theorem will care.

[@bigtiger1964]

Angles are named with Greek letters, in a triangle with the corner points A, B and C as α, β and γ. The sides are a, b and c. That’s 5th grade stuff. Naming the angles with Latin letters seems very odd to me. Thanks for your explications!

[@Grizzly01-vr4pn]

"Ughh, this maths is boring! When will we ever use this stuff in real life?"

[@lukeknowles5700]

When you use a compass to draw the circle you already know where the center is.

[@jimlassiter749]

GREEN PAPER....!?!?!? ARE YOU NUTS......!?!?!?

[@trien30]

😂An easier solution using origami based on my high school teacher's mantra of "Keep it simple, stupid.": Fold the circle into quarters, where the 2 diameters intersect would be the center of the circle.🎉

[@txtradesman527]

In smaller circles that will work. Larger circles, not so much.

[@adriansue8955]

the techniques described here are for use in The Real World, not On Paper. You can't fold a slab of concrete.

[@chuckh.2227]

Horrible green background color Very hard to see what your doing

[@creamwobbly]

1. Thales's _Thales_ is not a plural word, so it takes _'s_ like every other singular noun. 2. /TAL-ess/ It's a Greek name. Initial _th_ is plosive, not soft, and the _a_ is flat, like in _flat._ The _e_ is also flat, as in _bet._ Amazing how you can rack up so many errors in just one word, but there ya go. I guess how they do teacherin' in Taxes. And you spend almost nine minutes wittering on about something that's fully explained by the diagram on the Wikipedia article. Just amazing.

[@txtradesman527]

Oh Sweet Jesus! Just shut up already.

[@peterbaruxis2511]

I generally agree except "takes" in your comment sb "take's" - IDK what Grammarly say's, in this example "take's" isn't plural for take, it actuallt means "does take."

[@davidsorensen4400]

That is not Thales theorem You are using the central angle theorem

[@txtradesman527]

Yes it most certainly is.

[@davidsorensen4400]

@@txtradesman527 where are the parallel lines?

[@txtradesman527]

This isn’t the Proportionality Theorem.

[@davidsorensen4400]

@@txtradesman527 I know , it’s an application of the central angle theorem , twice .

[@txtradesman527]

It’s Thales’ Thereom, which is a special case of the Inscribed Angle Theorem.

[@brianomahoney7400]

The word is indentation. There is no such word as "indention!"

[@txtradesman527]

DEFINITION FOR INDENTION (1 OF 1) noun 1. The indenting of a line or lines in writing or printing. 2. The blank space left by indenting. 3. The act of indenting; state of being indented. 4. Archaic. An indentation or notch.

[@txtradesman527]

I would suggest that the next time you feel so inclined to attack someone based on your ”perceived” intelligence, you might want to have a clue about what you’re talking about first.

[@txtradesman527]

Shall I give you the definition of perceived as well, or do you think your simple minded brain can handle that big word?

[@saranevillerogueart9627]

Your idea too complicated. Mine easier