Slightly less elegantly one can demonstrate rotational and translational invariances. Translate by arbitrary d and we have the identity (a+d)^2 + (b+d)^2 + (c+d)^2 - (a+d)(b+d) - (b+d)(c+d)- (c+d)(a+d) = a^2 + b^2 + c^2 - ab - bc - ca = 0 For rotational and scaling invariance multiply each point by arbitrary w. We have the identity (wa)^2 + (wb)^2 + (wc)^2 - (wa)(wb) - (wb)(wc) - (wc)(wa) = w^2(a^2 + b^2 + c^2 - ab - bc - ca) = 0 So we can take the case a = 1, b=1/2 + i sqrt(3)/2, c=1/2 - i sqrt(3)/2 a^2 = a b^2 = c c^2 = b ab = b bc = a ca = c and the result follows

I think part of the reason some people are getting confused is that hearing the word “Pythagorean” and seeing the squares of a, b, and c might make it seem like those are the side lengths

It works for triangle made with 3 roots of unity. It works for any rotation/dilation about origin. Or multiplying by a constant. It works for any translation. Or adding a constant to each point.

Compile all these math procedures and processes into a book or website

😮

Neat! I always laugh at the "marker drop"

Hi Dr. Peyam! Pythagoras's mind would have exploded if he could have seen complex numbers and how they help relate geometry to algebra.

I think it would have been helpful to state when each of the two cases at the start applies. If I understood correctly, the first case happens when the points a, b, c occur in clockwise order in the triangle, whereas the second case happens when they occur in counterclockwise order. Either way, the transformation you need to perform to get from y to x is the same as the transformation needed to go from z to y, which is why x/y = y/z. I wonder if there is also a purely algebraic approach, starting from simply |a - b| = |a - c| = |b - c| and then working your way to the identity.

Very nice ;-)

Pythagoras would be proud!

Weird. If it's an equilateral triangle, all angles are 60 deg. a^2 + b^2 + c^2 = 3a^2 = 3b^2 = 3c^2 = .5a^2 -17.5b^2 + 2ab + pi*ac +(18-pi)bc

@drpeyam

Жыл бұрын

But a b c are not the length of the sides, just the coordinates of the points

@tw5718

Жыл бұрын

Ahh. I rewatched and I originally misunderstood the statement at the beginning about the vertices and using angles. I thought you were saying a b c were the angles. My brain went duh. Thanks for the reply and videos.

I don’t get it

@drpeyam

Жыл бұрын

?

@TheNetkrot

Жыл бұрын

@@drpeyam I don't get it either, I cant understand the vector addition you propose in the beginning, it seem odd or incorrect which I doubt coming from you. Thanks for the video anyway.

@drpeyam

Жыл бұрын

It’s correct, here a b c are coordinates of the vertices

@itssurge7946

Жыл бұрын

What in the world is a complex number

@drpeyam

Жыл бұрын

😂

Am I the only person who knows 4pi/3 is larger than pi the angle of a straight line. So there is some mistake

@drpeyam

Жыл бұрын

No mistake here

*WRONG.* You introduced a, b, c, x, y, z as complex numbers, and then treated them as real.