Proof using Complex Numbers: Ptolemy's Theorem

[@chengkaigoh5101]

Complex analysis is so interesting

[@renesperb]

A nice application , showing how useful complex numbers can be for treating geometric problems .

[@squeezy8414]

Just did this in a Cambridge STEP question, really cool stuff!

[@wannabeactuary01]

At 5:38 Let z = Rexp(im) and w = Texp(im) where R and T are moduli and m is the common argument of both complex numbers z/w = R/T = (R/T) exp(0) ie argument is zero as number is real and modulus R/T It might be obvious but never ever recall encountering this before!

[@DrBarker]

Not only is z/w real, but also z/w is a positive real number when they have the same argument. And z/w is negative iff arg(z) = -arg(w), so they lie on opposite sides of the same line through the origin.

[@KSignalEingang]

[Bulgarian accent] Verrry sophisticated. (If you know, you know)

[@josephb5417]

I know, love zvezdelina

[@Srinivasan.Answers]

Different approach!

[@user-rp6yq5rw3r]

Amazing bro!

[@user-gs4cq3uk2b]

I had never seen a vector proof of Ptolemy's theorem, so I tried it myself. (Proof by Vector) Let the position vectors of points B, C, and D be b, c, and d, respectively, with A as the starting point. AB･CD+AD･BC=AC･BD ⇔ |c-d||b|+|c-b||d|=|b-d||c| ① ∠ACD=∠ABD ⇒ c･(c-d)/(|c||c-d|)=b･(b-d)/(|b||b-d|) ② ∠ADB=∠ACB ⇒ d･(d-b)/(|d||d-b|)=c･(c-b)/(|c||c-b|) ③ ∠ADC+∠ABC=180° ⇒ d･(d-c)/(|d||d-c|)+b･(b-c)/(|b||b-c|)=0 ④ left side of ① = |b|(c-d)･(c-d)/|c-d|+|d|(c-b)･(c-b)/|c-b| =|b|((c-d)･c/|c-d|-(c-d)･d/|c-d|)+|d|((c-b)･c/|c-b|-(c-b)･b/|c-b|) (substitute ②④③ into the 1st, 2nd, and 3rd terms, respectively) =|b|((|c|/|b|)b･(b-d)/|b-d|-(|d|/|b|)b･(b-c)/|b-c|)+|d|((|c|/|d|)d･(d-b)/|d-b|-(c-b)･b/|c-b|) =|c|(b-d)･(b-d)/|b-d| =|c||b-d| =right side of ①

[@DrBarker]

Very neat, there are lots of ways to prove Ptolemy's theorem!

[@avyakthaachar2.718]

So neat👌👌

[@t.sambath]

Love ❣️❣️

[@VanNguyen-kx6gx]

Certified very long and wasted time in geometry. Use vectors to verify easily.

[@TechToppers]

I don't even think he went over a lot of geometry in the video aside from the fact that opposite angles are supplementary iff the quadrilateral is cyclic...