<instructions>Your goal is to analyze the following comment and estimate how highly it will be upvoted by the Hacker News community.</instructions><comment><author>jonahx</author><text>It&amp;#x27;s a very clever proof.&lt;p&gt;Agreed &amp;quot;using trigonometry&amp;quot; is potentially misleading.  After reading the proof, the only 2 senses in which &amp;quot;trignometry&amp;quot; is being used are:&lt;p&gt;1. The term &amp;quot;sin a&amp;quot; is used to denote the ratio opposite &amp;#x2F; hypotenuse.  But this can be considered a purely notational convenience.  They could have called it &amp;quot;foo a&amp;quot; and nothing would change, or they could have inlined the referred-to ratio everywhere.&lt;p&gt;2. The law of sines is required.  But the proof of this law [1] also boils down to nothing more than the ratio definition and some algebra.&lt;p&gt;So afaict no circular logic is being used, but at the same time it doesn&amp;#x27;t seem to be doing anything previously thought to be impossible, unless there was a previous belief that the law of sines could not be used in a proof, which would be a strange belief to hold.  I see it simply as a creative, unexpected proof.&lt;p&gt;[1] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.youtube.com&amp;#x2F;watch?v=4xO8xqLyEbA&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.youtube.com&amp;#x2F;watch?v=4xO8xqLyEbA&lt;/a&gt;</text><parent_chain><item><author>joe_the_user</author><text>I have read about this proof for a bit and this is the first write-up that gives the slightest details. The phrase &amp;quot;using trigonometry&amp;quot; is confusing. What they do is assume functions sine and cosine exist, as normally defined, as ratios of triangle values, without assuming these have the various Pythagorean-theorem derived properties. They then construct an infinite series of nested triangles and use the formula for the sum of geometric series&amp;#x27; to derive the length of the original triangle&amp;#x27;s hypotenuse. It certainly seems clever.&lt;p&gt;I&amp;#x27;m still confused what axioms they&amp;#x27;re effectively using relative to the usual Pythagorean theorem proofs - most of these use the formula for area of a right triangle and this seemingly doesn&amp;#x27;t. On the other hand, it seems an infinite construct would require things like the axiom of induction, which may or may not be included in axiom of axiomatic geometry.</text></item></parent_chain></comment><story><title>New Orleans teenagers found a new proof of the Pythagorean Theorem</title><url>https://keith-mcnulty.medium.com/heres-how-two-new-orleans-teenagers-found-a-new-proof-of-the-pythagorean-theorem-b4f6e7e9ea2d</url></story>
<instructions>Your goal is to analyze the following comment and estimate how highly it will be upvoted by the Hacker News community.</instructions><comment><author>scythe</author><text>Induction isn&amp;#x27;t so bad. I&amp;#x27;m not sure how they&amp;#x27;re getting away with the Law of Sines, though. The usual proof of LoS that I know is dependent on the existence of the circumcircle. But I don&amp;#x27;t know how to prove the existence of the circumcircle without dragging in a lot of geometry.  Or you can use the area formula, which makes the proof similar to other arguments that use the area formula.</text><parent_chain><item><author>joe_the_user</author><text>I have read about this proof for a bit and this is the first write-up that gives the slightest details. The phrase &amp;quot;using trigonometry&amp;quot; is confusing. What they do is assume functions sine and cosine exist, as normally defined, as ratios of triangle values, without assuming these have the various Pythagorean-theorem derived properties. They then construct an infinite series of nested triangles and use the formula for the sum of geometric series&amp;#x27; to derive the length of the original triangle&amp;#x27;s hypotenuse. It certainly seems clever.&lt;p&gt;I&amp;#x27;m still confused what axioms they&amp;#x27;re effectively using relative to the usual Pythagorean theorem proofs - most of these use the formula for area of a right triangle and this seemingly doesn&amp;#x27;t. On the other hand, it seems an infinite construct would require things like the axiom of induction, which may or may not be included in axiom of axiomatic geometry.</text></item></parent_chain></comment><story><title>New Orleans teenagers found a new proof of the Pythagorean Theorem</title><url>https://keith-mcnulty.medium.com/heres-how-two-new-orleans-teenagers-found-a-new-proof-of-the-pythagorean-theorem-b4f6e7e9ea2d</url></story>