1

I am working on an assignment proving Ptolemy Theorem using complex number, and I am looking at a textbook Complex Numbers and Geometry by Hahn. Here is what I am working at this moment:

THE PTOLEMY-EULER THEOREM:

For any four complex numbers a, b, c and d, the following identity is easy to verify:

(a-b)(c-d) + (a-d)(b-c) = (a-c)*(b-d).

By the triangle inequality, we obtain |a-b||c-d| + |a-d||b-c|> or =|a-c||b-d|.

Let us investigate when the inequality becomes an equality. In the case of triangle inequality, |z1 + z2|< or = |z1| + |z2|, equality holds iff z1/z2 is a positive number (provided z1*z2 not equals to zero). Thus we are looking for a condition to ensure that (a-b)(c-d)/(a-d)(b-c) is a positive real number.

But (a-b)(c-d)/(a-d)(b-c) is a positive real number, it is so iff

[(a-b)/(a-d)] / [(c-b)/(c-d)] is a a negative real number, it is so iff

arg{[(a-b)/(a-d)] / [(c-b)/(c-d)]} = arg{[(a-b)/(a-d)]} - arg{[(c-b)/(c-d)]} congruence to pi (mod 2pi)

If follows that a, b, c and d are cocyclic, ie., a, b, c and d are on the same circle or line and a and c are on the opposite sides of the chord joining b and d, which results in the alphabetical order (clockwise or counterclockwise.)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The textbook is very terse and does not have good explanation. Being not very well versed in complex analysis, here are my questions:

(a) What does (a-b), (a-d), etc. mean? Do they represent line ab, line ad, etc., in complex plane?

(b) I understand that "arg" means argument which is the angle theta in polar form. But what do arg{[(a-b)/(a-d)]}, arg{[(c-b)/(c-d)]} and arg{[(a-b)/(a-d)] / [(c-b)/(c-d)]} represent?

Any explanation would be very much appreciated and thank you for spending your precious time with this posting. Take care.

A.Magnus
  • 3,617
  • 6
  • 31
  • 48
  • $a,b,c,d$ are complex numbers, hence $b-a$ represents the vector $\vec{AB}$. For any complex number $z$, $z=|z|e^{i\theta}$ for a $\theta\in(-\pi,\pi]$. We say that $\theta=\operatorname{arg}(z)$. So, $\operatorname{arg}\frac{a-b}{a-d}$ represents the angle between $AB$ and $AD$. – Jack D'Aurizio Sep 15 '14 at 00:15
  • @JackD'Aurizio Thank you for your prompt response but that leads to another question: If arg{(a-b)/(a-d)} and arg {(c-b)/(c-d)} represent angle DAB and BCD respectively, these come down to (angle DAB)-(angle BCD) congruence to phi(mode 2phi), or (angle DAB)-(angle BCD) = 180, which certainly does not prove cocyclicity. As always, your help is very much appreciated. – A.Magnus Sep 15 '14 at 14:50
  • @JackD'Aurizio: Never mind about my question, I got it now. arg {(a−b)/(a−d)} does not necessary means the angle between AB and AD. Here, arg {(a−b)/(a−d)} = arg {(a−b)} - arg {(a−d)}, where arg {(a−b)} represents the vector's reference angle with the positive x-axis, the same for arg {(a−d)}. Thank you anyway for your help. – A.Magnus Sep 19 '14 at 18:07

0 Answers0