Here is my favorite. The number "1" lies on the real axis and on a unit circle centered on the origin of the complex plane. It has n nth unit roots u_(n,j) = e^(i*2πj/n) = cos(2πj/n) + i*sin(2πj/n), j = 0, 1, ..., n-1 where n > 1 and i = √(-1). All of these unit roots lie on that same unit circle. Prove that Σ[u_(n,j)]^k = 0, j = 0, 1, ..., n-1 for any integer k not an integer multiple of n, and where [u_(n,j)]^k means raise u_(n,j) to the kth power. [u_(n,j)]^k = e^(i*2πjk/n) = cos(2πjk/n) + i*sin(2πjk/n)
Deserves more views and likes. Unbelievably, I am 80 y/o and just today learned about the existence of the chord chord power theorem. I am quite sure that it never came up in high school geometry or college calculus.
'High school' geometry, in theory (pun), has so many wonderful theorems, that one could probably spend a whole year just doing them sans anything else.
As an engineer graduate, I can assure you they don't show up, I don't remember even actually using normal geometry in my college years, let alone circle theorems, but rn I'm circling RUclips looking up all circle theorems and collecting them together (gonna collect the written proofs later though "I have most of them already")
To what decimal place, (after 10 places the numbers change, should they?) I tested this in my CAD software and I got different numbers. A= 0.6211972 B= 0.8341628528 C= 0.2484103854 D= 2.0859821449 A*B= 0.51817962850337216 C*D= 0.51817962855212764446
I really love circle theorem proofs. Which is your favourite circle theorem?
Right angle in a semicircle is pretty good. Can be solved geometrically or with vectors. Also its one of he easier ones.
Well the triangles are similar but only congruent if the intersection point lies in the center of the circle.
Here is my favorite. The number "1" lies on the real axis and on a unit circle centered on the origin of the complex plane. It has n nth unit roots u_(n,j) = e^(i*2πj/n) = cos(2πj/n) + i*sin(2πj/n), j = 0, 1, ..., n-1 where n > 1 and i = √(-1). All of these unit roots lie on that same unit circle. Prove that Σ[u_(n,j)]^k = 0, j = 0, 1, ..., n-1 for any integer k not an integer multiple of n, and where [u_(n,j)]^k means raise u_(n,j) to the kth power. [u_(n,j)]^k = e^(i*2πjk/n) = cos(2πjk/n) + i*sin(2πjk/n)
Deserves more views and likes. Unbelievably, I am 80 y/o and just today learned about the existence of the chord chord power theorem. I am quite sure that it never came up in high school geometry or college calculus.
'High school' geometry, in theory (pun), has so many wonderful theorems, that one could probably spend a whole year just doing them sans anything else.
As an engineer graduate, I can assure you they don't show up, I don't remember even actually using normal geometry in my college years, let alone circle theorems, but rn I'm circling RUclips looking up all circle theorems and collecting them together (gonna collect the written proofs later though "I have most of them already")
Thank you professor for being TO THE POINT
This was an amazing explanation! Thank you for the enlightenment. 🙂
THANK YOU. No one explained this proof to me, but it makes so much sense now. Cheers!
Nice blackboard. I need one of those.
Learned it has to be subtended from the same arc bc of similar triangle relationship.I didnt under stand that earlyer in highschool.
Thankyou
Beautiful 😍
To what decimal place, (after 10 places the numbers change, should they?) I tested this in my CAD software and I got different numbers. A= 0.6211972 B= 0.8341628528 C= 0.2484103854 D= 2.0859821449 A*B= 0.51817962850337216 C*D= 0.51817962855212764446
The lengths should be exactly the same. Any errors are being introduced through rounding or some other interesting aspect of your software.