wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> integral inradius and circumradius
        
(Message started by: Christine on Nov 20^th, 2011, 12:43pm)

Title: integral inradius and circumradius
Post by Christine on Nov 20^th, 2011, 12:43pm

(1) For each integer n it is possible to construct an integer triangle whose inradius is n? How?

(2) Can we always have integral circumradius as well?

Title: Re: integral inradius and circumradius
Post by towr on Nov 20^th, 2011, 12:54pm

1) [hide]the triangle 3-4-5 has inradius 1, so any K-multiple would have radius K[/hide]

For 2) do you mean at the same time as having an integer triangle with integer inradius?

Title: Re: integral inradius and circumradius
Post by Christine on Nov 20^th, 2011, 1:06pm

Quote:

For 2) do you mean at the same time as having an integer triangle with integer inradius?

We may ask

(a) integral sides + integral circumradius
(b) integral sides + integer inradius + integer circumradius

Title: Re: integral inradius and circumradius
Post by Christine on Nov 20^th, 2011, 1:13pm

I just saw that it only works with Pythagorean triangle

I found these 2 theorems:

The inradius of a Pythagorean triangle is an integer.

The inradius of a Heronian triangle is a rational number.

But I don't know whether we always have integral circumradius as well (either at the same time or not)?