Subject: Re: [HM] Heron's formula
From: Lambrou Michael (lambrou@itia.math.uch.gr)
Date: Wed Feb 16 2000 - 08:35:41 EST
The following thread came from the HYACINTHOS discussion group, relating
to a geometrical proof of Heron's formula, as brought to our attention by
John Conway.
As HM members might be interested in the query I added, let me repeat it
here.
On Sun, 13 Feb 2000, John Conway wrote (on Heron's formula):
>
> Here's the proof, then. We start from the formulae
>
> DELTA = ro.so = ra.sa = rb.sb = rc.sc
>
> where the rx are the inradii and the sx the semiperimeters
> (+-a+-b+-c)/2, as usual.
>
> For these we have the usual easy proof -
>
> area ABC = area ABIo + Area BCIo + Area CAIo
>
> = a.ro/2 + b.ro/2 + c.ro/2 = ro.so
>
> for x = o, and for the others replace ro by ra,rb,rc
> and change the 1st,2nd,3rd sign.
>
> Now we look at the picture:
>
> Ia
> |\
> | \ Io
> | \ /|
> ra| \ / |ro
> | \ / |
> | \ / |
> ----X-----C--------Y------------------------A
> sb sc
>
> and see that since the angles of the two indicated triangles at C
> are 90 - C/2 and C/2 , these two triangles are similar, so that we
> have
> ro.ra = sb.sc,
>
> so that DELTA^2 = so.sa.ro.ra = so.sa.sb.sc.
>
> Somehow I'd never seen the remark about the two triangles before.
>
> I found it in Casey's Sequel to Euclid, of which, since Friday morning,
> I have a Xeroxed copy. What happened was that in Urbana I was introduced
> to their very nice mathematical librarian, unfortunately just about to
> retire, who asked me to see how long she'd take to find a given book. So,
> since I'd been thinking about it, of course I asked for Casey, and she
> duly produced it inside a minute. After reading through it throughout
> most of the night, I decided to Xerox it next morning. I'm glad I chose
> this particular copy, because bound in with it are some notes of its
> original owner, one F. William Wilson, some of which are quite useful.
(clip)
Wonderful. Let me add:
In a modern Greek edition of "Definitiones" and "Geometrica" of Heron of
Alexandria, the editor Kipouros says in a footnote, page xxxvi:
"There are 20 proofs of the formula, without the variations, as
classified using criteria of central arguments, by Botther Progr.
Petrischule, Leipniz 1909 (Tittel, RE, pages 1014/45)"
(Sorry readers, but I translate what I see). I presume RE is the eighth
volume of Pauly-Wissowa Realencyclopadie, where on pages 995ff there is
an article on Heron by Tittel.
Does any German speaker have access to the above? Can he tell Hyacinthos
(and HM) members about its content? I would be glad to know.
Michael Lambrou.
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