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antilos_weorsick t1_j2dgouh wrote

Reply to comment by mikesteane in ELI5: Angles in a semicircle by [deleted]

This assumes the triangle was already a right triangle. Circluar (pun intended) reasoning. Also it only explains those two specific triangles.

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mikesteane t1_j2dhqsq wrote

It does not assume that. Maybe it doesn't work for you, but I can see from this thought experiment, without needing the standard proof, that it can only be a rectangle.

If you need some interim steps between looking at the image and concluding that it is a rectangle (getting nearer to the proof that OP specifically said he didn't want) you can imagine trying to deform the quadrilateral to vary the angles while still maintaining the vertices on the circle, you can see that this is impossible. The quadrilateral is fixed. Looking at the symmetry again shows that all the angles are right angles.

Since the starting triangle was a random angle in a semi-circle, the visualisation applies to all such triangles.

Can you not see this without needing an algebraic or verbal proof?

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antilos_weorsick t1_j2dit0n wrote

The question is if you make a Thales triangle, is it a right triangle? You start by assuming that if you have a right triangle, then you flip it over the hypothenuse, and you get a rectangle. Well, of course you do, it was right triangle to begin with! If it wasn't a right triangle, and you did your construction, you wouldn't get a rectangle.

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mikesteane t1_j2dj9ac wrote

Maybe you can't see it, but I can.

I started with a triangle in a semi-circle as requested by OP and did not assume it was a right-angled triangle. However when completing the construction as I described, it is clear (to me at least) that it can only have been a right angle to start with. I filled out a couple of extra steps to help you complete the visualisation, but it seems you need the formal algebra. Maybe OP still needs the proof, but to me it is so obvious as not to need formal proof.

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antilos_weorsick t1_j2dnhlr wrote

Ok buddy, what happens if it isn't a right triangle? Never heard of a rhombus?

Also, you keep using the words "formal algebra". I don't think they mean what you think they mean.

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mikesteane t1_j2doafd wrote

That's my point; I'm not assuming it is a right triangle, but when the construction is drawn I (even if not you) can see that it must be one. I've even given you a couple of interim steps to seeing why this is so, but you don't seem to grasp it even then.

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na3than t1_j2e5wro wrote

You ARE assuming the original triangle is a right triangle. If you take a triangle ABC, rotate it 180° and translate it so that AC coincides with the original CA, you're guaranteed to get a PARALLELOGRAM. You're only guaranteed to get a rectangle if the original triangle was a right triangle.

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