# A014498 and JHC's extraversion

Antreas P. Hatzipolakis xpolakis at otenet.gr
Tue Jun 6 23:00:20 CEST 2000

```I wrote:

>ID Number: A014498
>Sequence:  1,2,3,3,6,5,4,10,5,12,7,15,14,6,15,20,9,21,7,18,28,11,8,21,
>           30,35,22,9,36,24,35,13,42,33,45,10,26,40,44,15,39,11,30,45,
>           55,56,30,63,52,12,33,66,17,63,65,72
>Name:      Varying radii of inscribed circles within primitive Pythagorean
>           triples as a function of increasing values of hypotenuse.
>
>___________________________________________________________________________
>Let's see if JHC's "extraversion"(**) applies here for the three exradii .

Someone asked me what Conway's extraversion is.

In my understanding, it is a metatheorem:
When you come across a theorem on the incircle/incenter/angle int.
bisectors, expect an analogous theorem on the excircles/excenters/
angle ext.bisectors.
(Conway, of course, express it in his own (algebraic) language, and I hope
that he will not second Johnson's words [see below my sig. file] to me :-)

The theorem on the incircle (used to define the integer sequence A014498) is:
The length of the radius of the circle inscribed in a Pythagorean
triangle is always an integer.
(C. Stanley Ogilvy - John T. Anderson: Excursions in Number Theory.
New York: Dover, 1988, p. 68. Originally publ. in 1966 by OUP)

Having in mind JHC's extraversion metatheorem, I found that the same
is true for the lengths of the radii of the three excircles.

Antreas

Sir, I have found you an argument. I am not obliged to find you
an understanding.
-- Samuel Johnson

```