5 ms·

Proving Heron’s formula by complex number (2013)

45 points by mehulashah 3y ago

kwertzzz 3y ago
Here is a "Proof of the Law of Cosines" by the same author: https://www.tandfonline.com/doi/pdf/10.4169/amer.math.monthly.121.02.149 https://www.tandfonline.com/doi/pdf/10.4169/amer.math.monthl... But this time not with complex numbers.
- mehulashah 3y ago
  So, I have to pay to read the proof? Not fun.
  - milesde 3y ago
    It's also on JSTOR: https://www.jstor.org/stable/10.4169/amer.math.monthly.121.02.149 https://www.jstor.org/stable/10.4169/amer.math.monthly.121.0... . No payment needed there (unless you've burned through your 100 free articles for this month).
    - ColinWright 3y ago
      I appreciate that, but it requires creating an account, and for people who have a gazillion other things to be doing, taking the time to create Yet Another Account on Yet Another System just for the one article is something most people will pass on. It may not have a financial cost, but it certainly has an opportunity cost, and contributes to "Account Fatigue". But loved the proof of Heron's formula, and I'm pleased to be able to say thank you in person, and make sure I give credit when I use it for the Math Club talks I do. Great work ... feel free to contact me if you are so inclined, or put an email address in your profile.
mehulashah 3y ago
I love it when imaginary numbers help simplify a complex explanation.
- BeetleB 3y ago
  Pun intended?
  - mehulashah 3y ago
    Took you long enough :P. Nonetheless, it’s also a beautiful proof.
- kkylin 3y ago
  https://xkcd.com/410/ https://xkcd.com/410/
  - mehulashah 3y ago
    ROFL. I hadn’t seen that one yet.
smacke 3y ago
Discussion on reddit from when I posted about this (more than 10 years ago now!): https://old.reddit.com/r/math/comments/144qc4/a_beautiful_proof_of_herons_formula/ https://old.reddit.com/r/math/comments/144qc4/a_beautiful_pr... Original pdf seems to have been taking down from AOPS but is still available in the internet archive: https://web.archive.org/web/20121018044426/https://artofproblemsolving.com/Resources/Papers/Heron.pdf https://web.archive.org/web/20121018044426/https://artofprob...
- mehulashah 3y ago
  Interesting, the proof seems to have originated as a submission to The Art of Problem Solving. I wish I had these site when I was in high school. I would have poured over them.
  - milesde 3y ago
    Close. I sent it to rrusczyk around the time it got accepted by the Monthly. I wanted him to see it because AoPS was playing a huge role in my mathematical education at the time, and was happy when he wanted to post it on the website.
    - mehulashah 3y ago
      This is a great story. Have you written it up somewhere? It’d be inspiring for the right high school kids.
  - pixelpoet 3y ago
    It's *pored over BTW
    - mehulashah 3y ago
      My bad. Typing fast.
bonzini 3y ago
I'm okay with conjuring (r+ix)(r+iy)(r+iz) out of thin air. I am also okay with adding an extra step at the bottom to make it a proof of Heron's formula, even though at its heart it's a proof of the radius of the incircle—because the former sounds way more cool. However, a complete proof really should explain the diagram at the top to properly define x, y and z. The idea, if I understand correctly, is to draw the altitudes of the three triangles AIB, BIC, CIA, and then show that the six resulting right triangles are congruent in pairs because they have one equal angle at A/B/C (the segments AI, BI, CI bisect the angles of the triangle ABC), and the hypotenuse in common. Not too complicated but even if a picture is worth a thousand words, it wouldn't hurt to have it written also in words. A subsequent step would prove that r is equal for all pairs of triangles, because it is a common side of triangles coming from different pairs. (As a bonus this proves that the incenter is indeed the center of the inscribed circle). Likewise, going from r+ix to u*e^iα is basically Pythagoras's theorem, but it would not hurt to write that explicitly. And it does not hurt to write "let s=(a+b+c)/2=x+y+z be the semiperimeter of the triangle" (x+y+z is not the definition of the semiperimeter!) and to only introduce it after x+y+z actually occurs somewhere. It's a really cool proof, and hey it was written by a high school student—but the proof writing is as important as the proof IMO.
- mehulashah 3y ago
  Interesting. Certainly, those small leaps are missing. But, that’s what made the proof fun for me. They were small enough that I could fill in the blanks. In some cases, it took a little thinking, but overall it was a nice way of approaching the construction.
- milesde 3y ago
  Hi! I wrote that proof back in 2007, half of my lifetime ago. It's neat to see it discussed again. My original writeup was longer and included several of the details that you mentioned. However, the editors of the American Mathematical Monthly condensed it, presumably so that it could be published as a "filler" that would take up less than a page. What you see is their rewrite of my work. In fairness to them, the stuff that's left out is well known to people who think about incircles and excircles and thus not really new. For instance, it's used in the proof of Heron's formula by similar triangles. There were some things that I would not have put that way on my own. For instance, at that point in my life, I had actually been trained not to use the word "clearly" like it's used here. But who was I, a high school student, to quibble with the editors of the American Mathematical Monthly? I was thrilled that they wanted to publish my work. No regrets.
prvc 3y ago
What kind of research goes into determining that the proof is a novel one, such that submitting it to a math journal becomes a promising prospect?
- milesde 3y ago
  I did a lot of Googling to see if it turned up, showed it to my math teachers and my mathematician parents, sent it off to the editors, and hoped for the best. 17 years have gone by without anybody pointing out an earlier instance of the proof to me, so it seems pretty safe at this point.

45 points by mehulashah 3y ago

Here is a "Proof of the Law of Cosines" by the same author: https://www.tandfonline.com/doi/pdf/10.4169/amer.math.monthly.121.02.149 https://www.tandfonline.com/doi/pdf/10.4169/amer.math.monthl... But this time not with complex numbers.

mehulashah 3y ago
So, I have to pay to read the proof? Not fun.
- milesde 3y ago
  It's also on JSTOR: https://www.jstor.org/stable/10.4169/amer.math.monthly.121.02.149 https://www.jstor.org/stable/10.4169/amer.math.monthly.121.0... . No payment needed there (unless you've burned through your 100 free articles for this month).
  - ColinWright 3y ago
    I appreciate that, but it requires creating an account, and for people who have a gazillion other things to be doing, taking the time to create Yet Another Account on Yet Another System just for the one article is something most people will pass on. It may not have a financial cost, but it certainly has an opportunity cost, and contributes to "Account Fatigue". But loved the proof of Heron's formula, and I'm pleased to be able to say thank you in person, and make sure I give credit when I use it for the Math Club talks I do. Great work ... feel free to contact me if you are so inclined, or put an email address in your profile.

mehulashah 3y ago

I love it when imaginary numbers help simplify a complex explanation.

BeetleB 3y ago
Pun intended?
- mehulashah 3y ago
  Took you long enough :P. Nonetheless, it’s also a beautiful proof.
kkylin 3y ago
https://xkcd.com/410/ https://xkcd.com/410/
- mehulashah 3y ago
  ROFL. I hadn’t seen that one yet.

smacke 3y ago

Discussion on reddit from when I posted about this (more than 10 years ago now!): https://old.reddit.com/r/math/comments/144qc4/a_beautiful_proof_of_herons_formula/ https://old.reddit.com/r/math/comments/144qc4/a_beautiful_pr... Original pdf seems to have been taking down from AOPS but is still available in the internet archive: https://web.archive.org/web/20121018044426/https://artofproblemsolving.com/Resources/Papers/Heron.pdf https://web.archive.org/web/20121018044426/https://artofprob...

mehulashah 3y ago
Interesting, the proof seems to have originated as a submission to The Art of Problem Solving. I wish I had these site when I was in high school. I would have poured over them.
- milesde 3y ago
  Close. I sent it to rrusczyk around the time it got accepted by the Monthly. I wanted him to see it because AoPS was playing a huge role in my mathematical education at the time, and was happy when he wanted to post it on the website.
  - mehulashah 3y ago
    This is a great story. Have you written it up somewhere? It’d be inspiring for the right high school kids.
- pixelpoet 3y ago
  It's *pored over BTW
  - mehulashah 3y ago
    My bad. Typing fast.

bonzini 3y ago

I'm okay with conjuring (r+ix)(r+iy)(r+iz) out of thin air. I am also okay with adding an extra step at the bottom to make it a proof of Heron's formula, even though at its heart it's a proof of the radius of the incircle—because the former sounds way more cool. However, a complete proof really should explain the diagram at the top to properly define x, y and z. The idea, if I understand correctly, is to draw the altitudes of the three triangles AIB, BIC, CIA, and then show that the six resulting right triangles are congruent in pairs because they have one equal angle at A/B/C (the segments AI, BI, CI bisect the angles of the triangle ABC), and the hypotenuse in common. Not too complicated but even if a picture is worth a thousand words, it wouldn't hurt to have it written also in words. A subsequent step would prove that r is equal for all pairs of triangles, because it is a common side of triangles coming from different pairs. (As a bonus this proves that the incenter is indeed the center of the inscribed circle). Likewise, going from r+ix to u*e^iα is basically Pythagoras's theorem, but it would not hurt to write that explicitly. And it does not hurt to write "let s=(a+b+c)/2=x+y+z be the semiperimeter of the triangle" (x+y+z is not the definition of the semiperimeter!) and to only introduce it after x+y+z actually occurs somewhere. It's a really cool proof, and hey it was written by a high school student—but the proof writing is as important as the proof IMO.

mehulashah 3y ago
Interesting. Certainly, those small leaps are missing. But, that’s what made the proof fun for me. They were small enough that I could fill in the blanks. In some cases, it took a little thinking, but overall it was a nice way of approaching the construction.
milesde 3y ago
Hi! I wrote that proof back in 2007, half of my lifetime ago. It's neat to see it discussed again. My original writeup was longer and included several of the details that you mentioned. However, the editors of the American Mathematical Monthly condensed it, presumably so that it could be published as a "filler" that would take up less than a page. What you see is their rewrite of my work. In fairness to them, the stuff that's left out is well known to people who think about incircles and excircles and thus not really new. For instance, it's used in the proof of Heron's formula by similar triangles. There were some things that I would not have put that way on my own. For instance, at that point in my life, I had actually been trained not to use the word "clearly" like it's used here. But who was I, a high school student, to quibble with the editors of the American Mathematical Monthly? I was thrilled that they wanted to publish my work. No regrets.

prvc 3y ago

What kind of research goes into determining that the proof is a novel one, such that submitting it to a math journal becomes a promising prospect?

milesde 3y ago
I did a lot of Googling to see if it turned up, showed it to my math teachers and my mathematician parents, sent it off to the editors, and hoped for the best. 17 years have gone by without anybody pointing out an earlier instance of the proof to me, so it seems pretty safe at this point.