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!

[Myrus_MBG]

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.

[gardening_vibes]

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

[sumaitarimjhim]

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

[atharvagarwal]

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

[srijaghosh]

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

[mike_the_tutor]

Excellent video. Thank you for sharing.

[DragonKidPlaysMC]

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

[ujjwaldhawan]

dont knw why good youtubers have so less

[az]

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.

[utkarshkathuria]

How can I prove using Complex numbers?

[sathwik]

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

[debarshimukherjee]

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

[brokelama]

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

[_Abdur_Rob]

Which theorem was invented first Pythagoras or Ptolemy??

[TechToppers]

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