Complex plane

Scorpius · P versus NP Offline Joined: 12 Jun 2009 Posts: 21

Is there any way to obtain a nice formula in the complex plane for the Gergonne or Nagel points, for example by choosing a non-usual origin or smt. like that?

**Posted:** Sat Sep 12, 2009 5:33 am

leshik · Poincare Conjecture Offline Joined: 03 Aug 2009 Posts: 156

What is the definition of ''usual'' origin? The most suitable way (if I remember correctly this part of school math Smile

) to choose incenter as origin or do you need some ''nicer'' formulas?

**Posted:** Sat Sep 12, 2009 5:46 am

Bugi

Usual origin is circumcenter, as then $|a|=|b|=|c|=1$

**Posted:** Sat Sep 12, 2009 1:44 pm

leshik · Poincare Conjecture Offline Joined: 03 Aug 2009 Posts: 156

OK, then I gave 'unusual' one Very Happy

To be serious, this is natural choice in most cases, when the problem related to incircle of the triangle.

**Posted:** Sat Sep 12, 2009 4:46 pm

Scorpius · P versus NP Offline Joined: 12 Jun 2009 Posts: 21

leshik,
I asked for a method to express the Nagel or the Gergonne point by a neat formula. The incenter is generally an often choice, but calcuting the asked points give us incredible difficulties in that case. That's why I created this topic, in the hope to find an answer.

**Posted:** Sun Sep 13, 2009 8:16 am

Bugi

Read Andreescu's Complex Numbers from A to Z:

Nagel point:
$\alpha,\beta,\gamma$ - lengths of the sides of the triangle
$a,b,c$ - complex numbers of vertexes
$s$ - halfperimeter

$z_N=\sum_{cyclic} a\left(1-\frac{\alpha}{s}\right)$

**Posted:** Sun Sep 13, 2009 9:31 am

Scorpius · P versus NP Offline Joined: 12 Jun 2009 Posts: 21

Thank you very much, Bugi.

**Posted:** Mon Sep 14, 2009 6:58 am