How to Prove Ptolemy's Theorem for Cyclic Quadrilaterals

Ptolemy's Theorem relates the diagonals of a quadrilateral inscribed in a circle to its side lengths. We give a proof of this theorem together with an application to a classical geometry theorem.
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[@Szynkaa]

there is also slightly extended version of it- namely we have inequality AD*BC+AB*DC>= AC*BD with equality iff quadrilateral is cyclic.

[@MuPrimeMath]

Adding the point K on AC is such an interesting step! I hope I can learn how to do geometric proofs like these at some point!

[@ProfOmarMath]

It's an interesting creative step. Maybe too magical!

[@myrus5722]

I always love those "magic line" proofs :) -- They really embody the creativity and ingenuity in proof geometry (which is often seen as the boring, rigorous way to do things) that is often missed by those that don't study it as deeply (that can be said for almost all subjects, too).
Thank you for this pretty cool proof. There are certainly more beautiful proofs, like the inversion one, but I still really like the approach you took. You seem more into mathematics than a lot of other people I've seen with similar channels... It doesn't feel like you're parroting a website and dryly restating the proof, but actually sharing your love of math with the audience.

[@ProfOmarMath]

Thanks Myrus! I really do love the stuff and wish I could share so much more

[@gardening_vibes]

Thank you for this. I was searching for proof of Ptolemy's theorem.

[@ProfOmarMath]

Definitely!

[@atharvagarwal6412]

Loved it! This channel is a gem. Here from blackpenredpen's video :D

[@ProfOmarMath]

Thanks!

[@srijaghosh9723]

Thank you sir for explaining in such an easy way 😊🙏

[@fernandoadrianromeroalvare2612]

Excellent explanation Omar!

[@mike_the_tutor1166]

Excellent video. Thank you for sharing.

[@ProfOmarMath]

Thank you!

[@sumaitarimjhim3926]

thanks very much!! I was really struggling with the proof I have in my book, yours is much easier to absorb.

[@az8887]

you should do more things like this. I am personally a great fan of planimetry, but sometimes feel a bit too lazy to read the proofs myself.

[@ProfOmarMath]

More coming 😊

[@krishshah3974]

Noice! A slow clear explanation!

[@ProfOmarMath]

Thanks!

[@DragonKidPlaysMC]

Can you olympiad techniques for euclidean geometry like inversion and barycentric coordinates?

[@ProfOmarMath]

Hi Drag. I typically focus on undergrad math material but I have some interesting techniques that apply to geometry coming quite soon that are quite surprising! Did you think about this from the inversion proof of Ptolemy?

[@rksharma7132]

Cristal clear 👌

[@sathwik5460]

thank you .

[@ProfOmarMath]

Definitely!

[@hvcreator2380]

THANK U SOO MUCH ❤

[@TechToppers]

This proof looks like magic. Really. Inversion looks much more intuitive.

[@ProfOmarMath]

Truth. Inversion takes a lot of set up. There is a nice numberphile about it actually

[@TechToppers]

@@ProfOmarMath
Yeah. I have a small request. Can you make a video on what's the idea of Circular Inversion, because that definition looks rather strange on first look but solves circle related problems in a jiffy.

[@sathwik5460]

can you please advice me some good book which has angle chasing problems ?

[@ProfOmarMath]

Hmmmm I actually can’t think of a specific outstanding one!

[@susancheng4782]

great! thank you! very enlightening!

[@ProfOmarMath]

You're very welcome!

[@brokelama7191]

Thank you so much for the explanation. :D. It really helped.

[@ProfOmarMath]

Awesome!

[@thequantmentor]

Very good video!! Keep going :)

[@utkarshkathuria2931]

How can I prove using Complex numbers?

[@王剛-m7n]

Great！ Beautiful！

[@SuperYoonHo]

subbed

[@015_Abdur_Rob]

Which theorem was invented first Pythagoras or Ptolemy??

[@ProfOmarMath]

I'm not sure!

[@radheya]

Nice explanation

[@ProfOmarMath]

Thanks!

[@debarshimukherjee8755]

Sir how did you get the idea to construct the angle bisector? Or you knew it beforehand?

[@ProfOmarMath]

This is indeed something I had seen before!

[@atharavyadav2979]

@@ProfOmarMath plz prove it

[@SeanBenson23]

@@atharavyadav2979 dude it doesn’t need to be proved. You’re allowed the make that construction. Use a protractor and a straightedge.

[@atharavyadav2979]

@@SeanBenson23 my friend i was asking therotical proof . yeah always we can prove it from construction not only that but even ptolmey theorem can proved by construction

[@anas8183]

نايس

[@ujjwaldhawan281]

dont knw why good youtubers have so less subs.....

[@ProfOmarMath]

Thanks Ujjwal. Can’t control the algorithm!

[@mrshockgod4476]

good vid

[@ProfOmarMath]

Thanks!

[@yashraj8028]

❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤👍👍👍👍👍👍👍👍👍👍👍👍👍

[@pranshu05]

2:05 Khabi Lame moment