4 ms·

> Proving that a parallelogram with equal diagonals is a rectangle is an exercise in itself; I’d prove it through Thales’ theorem, myself.. IIRC you prove it v

by lloeki 2y ago

> Proving that a parallelogram with equal diagonals is a rectangle is an exercise in itself; I’d prove it through Thales’ theorem, myself.. IIRC you prove it via angles. a) The sum of angles of any triangle is pi. b) The sum of angles of any quadrilateral is 2*pi. c) Since you rotate the triangle, opposite angles end up adding together. Because of the above, a bit of reasoning around symmetry shows that when diagonals are equal they can only add to pi/2 and that the other ones can only be pi/2. Any other angle leads to a contradiction. I mentioned angle values but (again IIRC) this can all be proven with compass and ruler.

vidarh 2y ago
As the article mentions, in this particular case it's simpler: You have a point on the triangle that is also on the circle. Now you rotate the triangle 180 degrees, with the point following the circle. The point inherently must end up exactly opposite the original point, and so the diagonal formed between the old and new point must be the diameter of the circle.
- lloeki 2y ago
  This proves that the parallelogram diagonals, both being diameters of the same circle, are equal, akho was pointing out that there's a leap from that to "the parallelogram is a rectangle", which can be proven with Thales. But then, the article is about laying out a proof of Thales so if it's using a lemma that leveraged Thales itself that's circular logic right there and so we're in trouble. One other way to prove Thales is by computing angles, but then you've got yourself a proven Thales so the point of the article is moot.
  - vidarh 2y ago
    I didn't read that very carefully, but proving that a parallelogram with equal diagonals is a rectangle is obvious enough via triangle congruency that I suspect they were just taught it without the proof. It's not a particularly interesting detail.
vlovich123 2y ago
Um… small correction. Sum of angles in triangle is 180 not pi (right-angle triangle is 90 + 45 + 45). Similarly for quadrilateral it’s 360 not 180 (a canonical being rectangles which is 4 90 degree angles).
- akho 2y ago
  What do you think pi is.
  - vlovich123 2y ago
    Apparently HN doesn't support the facepalm emoji.
- deleted 2y ago
  [deleted]
akho 2y ago
“Standard” Thales proof (via two isosceles triangles) is simpler than this.
- lloeki 2y ago
  It is, but Thales is what this article attempts to prove (a). By the point you've proven Thales with isosceles (b), proving that a parallelogram with equal diagonals is a rectangle thanks to Thales(b) in order to go forward along the Thales(a) proof is circular logic.
jasomill 2y ago
In case (a) and (b) aren't immediately obvious: The net change in direction around a triangle, quadrilateral, or any simple closed curve is a full rotation, i.e., 2π. The sum of the interior angle and the signed change in direction around a vertex of a polygon is π (consider the case of a "vertex" with interior angle π). So for a polygon with interior angles θ₁, θ₂, …, θₙ, 2π = (π - θ₁) + (π - θ₂) + ⋯ + (π - θₙ), therefore θ₁ + θ₂ + ⋯ + θₙ = nπ - 2π = (n-2)π.

by lloeki 2y ago

vidarh 2y ago

As the article mentions, in this particular case it's simpler: You have a point on the triangle that is also on the circle. Now you rotate the triangle 180 degrees, with the point following the circle. The point inherently must end up exactly opposite the original point, and so the diagonal formed between the old and new point must be the diameter of the circle.

lloeki 2y ago
This proves that the parallelogram diagonals, both being diameters of the same circle, are equal, akho was pointing out that there's a leap from that to "the parallelogram is a rectangle", which can be proven with Thales. But then, the article is about laying out a proof of Thales so if it's using a lemma that leveraged Thales itself that's circular logic right there and so we're in trouble. One other way to prove Thales is by computing angles, but then you've got yourself a proven Thales so the point of the article is moot.
- vidarh 2y ago
  I didn't read that very carefully, but proving that a parallelogram with equal diagonals is a rectangle is obvious enough via triangle congruency that I suspect they were just taught it without the proof. It's not a particularly interesting detail.

vlovich123 2y ago

Um… small correction. Sum of angles in triangle is 180 not pi (right-angle triangle is 90 + 45 + 45). Similarly for quadrilateral it’s 360 not 180 (a canonical being rectangles which is 4 90 degree angles).

akho 2y ago
What do you think pi is.
- vlovich123 2y ago
  Apparently HN doesn't support the facepalm emoji.
deleted 2y ago
[deleted]

akho 2y ago

“Standard” Thales proof (via two isosceles triangles) is simpler than this.

lloeki 2y ago
It is, but Thales is what this article attempts to prove (a). By the point you've proven Thales with isosceles (b), proving that a parallelogram with equal diagonals is a rectangle thanks to Thales(b) in order to go forward along the Thales(a) proof is circular logic.

jasomill 2y ago

In case (a) and (b) aren't immediately obvious: The net change in direction around a triangle, quadrilateral, or any simple closed curve is a full rotation, i.e., 2π. The sum of the interior angle and the signed change in direction around a vertex of a polygon is π (consider the case of a "vertex" with interior angle π). So for a polygon with interior angles θ₁, θ₂, …, θₙ, 2π = (π - θ₁) + (π - θ₂) + ⋯ + (π - θₙ), therefore θ₁ + θ₂ + ⋯ + θₙ = nπ - 2π = (n-2)π.