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

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.

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.

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.")

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.

Brilliantly explained and to the point. Excellent! Thanks.

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

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.

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.

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.

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.

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.

That's pretty cool 👍

Love the story.

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

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.

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.

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.

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

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

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

You just be a Millwright LOL great video

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

Very useful! Now are there 2 thales theorems?

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

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!!!

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

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.

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

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

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!

Black pencil on green paper makes poor vision.

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

😂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.🎉

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

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

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

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.

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

Horrible green background color Very hard to see what your doing

Your idea too complicated. Mine easier