4 ms·

> Since the triangle got turned completely around, the sides of the box must be parallel, so it makes a parallelogram. But it can't be a slanted box because bot

by t_mann 2y ago

> Since the triangle got turned completely around, the sides of the box must be parallel, so it makes a parallelogram. But it can't be a slanted box because both of its diagonals are diameters of the circle, so they're equal, which means it must be an actual rectangle. I'd be careful with such "visual proofs", even more so if accompanied by such handwavy reasoning. Eg, do we know that both diagonals are diameters? Do we know that a parallelogram with equal diagonals is a rectangle? While in this case things do work out nicely, I'd say this is almost more luck than a real proof - it's easy to mistakenly "prove" stuff like Pi=4 with similar reasoning. I believe 3B1B even has a video on the topic.

pringk02 2y ago
> do we know that both diagonals are diameters This must be true, because the diagonals are both straight lines that go through the centre and are bound by the edges, so it follows they must be equal to the diameter of the circle by definition. > Do we know that a parallelogram with equal diagonals is a rectangle? As another commenter points out, this is a theorem you can reach for, but proving it by itself is a bit more of a task.
- HarHarVeryFunny 2y ago
  > This must be true, because the diagonals are both straight lines that go through the centre How do we know the 2nd diagonal goes through the center ? Is it because of the construction by rotation ?
  - surajms 2y ago
    Yes. So, here when we rotate the triangle, we are essentially rotating each of the endpoints. For each endpoint, we rotate it by 180 degrees around the line segment joining the endpoint and the center. This by definition will result in a new position for each endpoint that creates a chord (as the two endpoints lie on the circle) and passes through the center (we rotated around it). A chord that passes through the center is by definition a diameter.
    - HarHarVeryFunny 2y ago
      Thanks. Thinking about it, another way of looking at it is to construct the rotated triangle by drawing a line from each point through the center to where it intersects with the circle on the other side. This is obviously a 180' rotation, but by construction we explicitly know the diagonal goes through the center.
  - lupire 2y ago
    It maybe more clear if you visualize the reflection as a pair of perpendicular reflections, first across the diameter (which is also across the center) and then internally reflecting the diameter (which is again also across the center.) Two reflections with a common fixed point make a rotation around that fixed point (angle of reflection is double the angle between the reflection axes.). Two perpendicular reflections make a 180 degree rotation around the intersection of the axes of rotation.
trueismywork 2y ago
There's nothing handwaving or luck about arguments by symmetry. Your pi=4 example has more defects than defects in symmetry arguments.
tech_ken 2y ago
I think the point is less to provide a completely proper proof of Thales Theorem, and more to demonstrate the fundamental principle of what a proof is (an argument to back-up a seemingly intractable statement), and how one might construct one (use concepts which we already understand, ex. rectangles, to create some plausible reasoning). Yes it involves some bad habits (relying primarily on visual intuition), but you've got to start somewhere. Moreover, the deficiencies of the example become the motivators for the next example ("so in the last example we did X, but that has problem A, so now we try Y").
- t_mann 2y ago
  ok, but if it's for pedagogical purposes, then it's even more important to point out that you're not done here (which the article misses). also, but this is purely personal preference, I don't think geometrical constructions are a good way to introduce someone to proofs. it's easy to fall into traps of circular reasoning (as in this example) or outright wrong arguments, it's not clear when you've done enough work, and it doesn't generalize well to other problems (what is the general strategy here - "draw more stuff and hope you see something"?). personally, I'd much rather have an introduction to proofs eg through something like Induction - it's very clear when you're 'done', there's no debate about whether it's a 'real' proof, much less risk of falling into traps, it's closer to university-level math and, most importantly, it's a versatile general tool that is easy to apply to new problems (and easy to see where it can be applied)
  - lupire 2y ago
    Geometrical constructions are valid if you use axioms. You can misuse induction (all horses are the same color, heap of sand cannot exist) and algebra (1 = 2 via division by zero)

by t_mann 2y ago

pringk02 2y ago

> do we know that both diagonals are diameters This must be true, because the diagonals are both straight lines that go through the centre and are bound by the edges, so it follows they must be equal to the diameter of the circle by definition. > Do we know that a parallelogram with equal diagonals is a rectangle? As another commenter points out, this is a theorem you can reach for, but proving it by itself is a bit more of a task.

HarHarVeryFunny 2y ago
> This must be true, because the diagonals are both straight lines that go through the centre How do we know the 2nd diagonal goes through the center ? Is it because of the construction by rotation ?
- surajms 2y ago
  Yes. So, here when we rotate the triangle, we are essentially rotating each of the endpoints. For each endpoint, we rotate it by 180 degrees around the line segment joining the endpoint and the center. This by definition will result in a new position for each endpoint that creates a chord (as the two endpoints lie on the circle) and passes through the center (we rotated around it). A chord that passes through the center is by definition a diameter.
  - HarHarVeryFunny 2y ago
    Thanks. Thinking about it, another way of looking at it is to construct the rotated triangle by drawing a line from each point through the center to where it intersects with the circle on the other side. This is obviously a 180' rotation, but by construction we explicitly know the diagonal goes through the center.
- lupire 2y ago
  It maybe more clear if you visualize the reflection as a pair of perpendicular reflections, first across the diameter (which is also across the center) and then internally reflecting the diameter (which is again also across the center.) Two reflections with a common fixed point make a rotation around that fixed point (angle of reflection is double the angle between the reflection axes.). Two perpendicular reflections make a 180 degree rotation around the intersection of the axes of rotation.

trueismywork 2y ago

There's nothing handwaving or luck about arguments by symmetry. Your pi=4 example has more defects than defects in symmetry arguments.

tech_ken 2y ago

I think the point is less to provide a completely proper proof of Thales Theorem, and more to demonstrate the fundamental principle of what a proof is (an argument to back-up a seemingly intractable statement), and how one might construct one (use concepts which we already understand, ex. rectangles, to create some plausible reasoning). Yes it involves some bad habits (relying primarily on visual intuition), but you've got to start somewhere. Moreover, the deficiencies of the example become the motivators for the next example ("so in the last example we did X, but that has problem A, so now we try Y").

t_mann 2y ago
ok, but if it's for pedagogical purposes, then it's even more important to point out that you're not done here (which the article misses). also, but this is purely personal preference, I don't think geometrical constructions are a good way to introduce someone to proofs. it's easy to fall into traps of circular reasoning (as in this example) or outright wrong arguments, it's not clear when you've done enough work, and it doesn't generalize well to other problems (what is the general strategy here - "draw more stuff and hope you see something"?). personally, I'd much rather have an introduction to proofs eg through something like Induction - it's very clear when you're 'done', there's no debate about whether it's a 'real' proof, much less risk of falling into traps, it's closer to university-level math and, most importantly, it's a versatile general tool that is easy to apply to new problems (and easy to see where it can be applied)
- lupire 2y ago
  Geometrical constructions are valid if you use axioms. You can misuse induction (all horses are the same color, heap of sand cannot exist) and algebra (1 = 2 via division by zero)