Heron's Formula, but Proven Elegantly

A beautiful and elegant mathematical proof to Heron's formula which uses the incircle and excircles of a triangle. This proof is not very well known, however, it is much more intuitive than the mainstream, algebraic proofs.
Thank you to Mr. Siwanowicz for showing me the proof on that fateful day two and a half years ago, and thank you to my friend asianpear for helping me so much with the video.
For more stuff, check out my website: kiraprograms.com
0:00 Introduction
1:07 Find lengths from the incircle
3:14 Find lengths from the excircle
4:11 Use similar triangles to solve for the inradius
7:13 Use the inradius for the area
8:20 Summary and conclusion

[@cnidariantide4207]

Thanks for the proof, fellow mathophile, I really enjoyed it. Splitting the triangle into six and plugging in the incircle radius to find the area is a classic visual proof and should be available to everyone. Once you've done that, it only remains to derive the incircle radius (the "inradius"), but I find that's the part which is harder to explain to youngsters and novices. I'll have to think about it some more and work through my Secret Life of Triangles book for fresh ideas. Certainly, no one should be forced to memorize Heron's formula without the chance to enjoy this very beautiful and elegant explanation of why it works! By the way, did you know that Heron's formula is a specific case of a formula found by Brahmagupta? Brahmagupta showed that if you draw a circle, and connect any four points on the circle to get a four-sided shape (an "inscribed quadrilateral") then you have, for its area, the formula S * S = (s-a)(s-b)(s-c)(s-d), where S is the area, s is the semiperimeter and a, b, c, d are the four sides. Heron's formula is the special case as one side shrinks to zero-so your '(s-d)' turns into plain 's' without the 'd' as your quadrilateral turns into a triangle. This works for all triangles because any triangle can be inscribed in a circle. Ah, I never get tired of geometry. Thanks again for the vid! Keep 'em coming!

[@lynnkim1217]

This is an excellent teaching video! Clear voice, clear illustrations and highly entertaining!

[@rkmoitra123]

This is the only proof which keeps the necessity of the quantity 's' to the forefront. In all other proofs 's' appears as an artefact, never central to the structure of the proof. Brilliant!

[@tasiemiecuzbrojony]

Piękny dowód, pierwszy raz widzę takie cuda... dziękuję bardzo 👍💯

[@JanSiwanowicz]

This is the best video in the history of youtube videos.

[@mathophilia]

Thanks! That means a lot coming from you :D

[@genekim331]

Very well done! You explained the proof (I hate proofs) in an easy to understand way that makes sense-all without trig! Keep them coming!

[@YoungCarWiz]

I see potential in this channel!

[@elizabethlee8341]

Loved this so much! Such an elegant and entertaining proof :D

[@suzettejansen9808]

Your first video.Totally awesome.With all that energy,clarity and precision you will achieve wonders.

[@comic4relief]

Rather interesting. I would like to suggest a small consideration: that you not say '90 degrees'. For in pure geometry degrees are not used. Two lines are perpendicular, or meet in a right angle, or some such. Degrees are for trigonometry or algebra which you said you are avoiding.

[@Jack_Callcott_AU]

This is a marvellous proof.. The other proofs of Heron's formula have a lot of tedious algebra. I downloaded it for safekeeping. 👍🔆

[@leonhorvat]

Conceptually beautiful Road to Brahmagupta is now paved

[@ashiqahmed1357]

Some times we have Semiperimetere less than the side of that triangle what will do than

[@unbelievable961]

Don't take this seriously 😅 but this video is more a comedy video than a maths video 😂😂😂...∞ Very nice job 👍 , keep making video...∞

[@benjobes776]

Brilliant and entertaining...nicely done!

[@eduardoteixeira869]

Very nice. Thank you

[@bennnlewis]

Great video, I really enjoyed it!

[@yoitsvic9720]

Great vid keep up the hard work Got CRAZY SKILLs ! dont give up stay with it !! i make vids too i seen you on discord i know it could get tuff sometimes but stay with it !! Left A LIKE !!✌👍✌✌

[@anyam6916]

Cool video!

[@YoungCarWiz]

Wow 17 views for a first video. That discord group really does help 😲

[@kennethren504]

Nice Effort!

[@mordechaimalka]

This is interesting proof of the formula but i have one problem at the beginning of the presentation , You show the formula but you don’t say what is s , that is : s=(a+b+c)/2

[@andrewwilson3395]

That was a great proof. Thank you.

[@vibrate1978]

The most concise proof by incircle and excircle.

[@elisabethgonzalez3817]

I can see it is supposed to make sense but still am a tad lost about half way through. Looks accurate though!

[@mrminer071166]

Hi mathophilia, You know what would be REALLY CUTE? Is to do an animation showing the three incircle triangles unrolling like beads on a string, so that all 3 rotated to standard A=bh/2, "bases on the ground" orientation, so the a+b+c was laid out along the "baseline," and then draw another line showing that ALL THREE triangles, have the same height, r. (Sketch a horizontal line going through all the apexes.) Then it's pretty obvious that the area would be r (a+b+c) /2. But then. . . . what's the geometric significance of the formula you give for r? I mean, what's the cool animation of that formula? The video you have here is VERY, VERY good . . . . but it's not quite "liquid math," where it looks totally obvious. That's what I always look for, videos which make the whole class go OH!!!

[@forthrightgambitia1032]

Yup this is begging for a manim treatment. Maybe someone can do it and use the voice over the top.

[@vibrate1978]

🤣🤣🤣 Cute, and adorable solution. Incredible. Great.👍🏻👍🏻👍🏻

[@narimatsuura1747]

Love your video!

[@charlesbromberick4247]

I might say Hero´s formula has not been considered as good math for at least 16 years - afterall, any theorem that depends on a "semi-perimeter" must be bunk.

[@learningisfun889]

Excellent work

[@wintutorials2282]

Thanks!!!

[@KLiCuk1]

superb 👍

[@ShahzadaIbrahim-r2k]

Love you ❤

[@n8dev]

sick

[@theimperator1327]

Channel still alive?

[@arsicjovan9171]

Epic! Thank you!

[@shubhamkamble5831]

Ok

[@the_nuwarrior]

NICE

[@ashiqahmed1357]

Please help

[@mani7619]

Intelligent

[@hiidostuff6786]

now this is epic

[@Shuutuu]

I love ❤

[@ericsimpson2751]

love it

[@klausolekristiansen2960]

This is the most complicated proof of this formula I have seen.

[@carmenreacts1861]

:)

[@carmenreacts1861]

MWAHAHAHA

[@mathophilia]

Ikr that part so didn't work xD

[@johnnisshansen]

More simple ? I Disagree