Geometric Proof of Heron's Formula

Date: 01/25/2000 at 17:43:50
From: Joey Lloyd
Subject: How do I prove geometrically Hero(n)'s formula?
How do I prove Hero(n)'s formula using a circle with center P and
radius R inscribed in triangle ABC?
Thank you so much.

Date: 01/26/2000 at 12:00:45
From: Doctor Floor
Subject: Re: How do I prove geometrically Hero(n)'s formula?
Hi, Joey,
Thanks for your question.
I know a method to prove Heron's formula geometrically with the help
of the incircle of the triangle, but in this method I also use one of
the excircles. I learned this method from Paul Yiu of Florida
Atlantic University.
First, I have made a picture for you of the triangles and the two
circles. Note that I have changed names: the incircle has center I and
radius r (while the excircle (opposite to A) has center I' and radius
r') instead of your center P and radius R.
In the following, the sides of the triangle are written as a = BC,
b = AC, and c = AB. I use s = (a+b+c)/2 for the semiperimeter.
1. From the fact that two tangents to a circle are congruent, we see
that AE = AG, CG = CF and BE = BF. So, for instance:
AE+EB+CG = c+CG = s
and thus CG = s-c.
In the same way:
CF = s-c
AG = AE = s-a
BE = BF = s-b
2. Again from the fact that two tangents to a circle are congruent we
see that AE' = AG, BE' = BJ and CG' = CJ. This gives us:
AG'+AE' = AB+BJ+CJ+AC = 2s
And we can conclude that AG' = AE' = s. And, for instance,
BE' = s-c.
3. Both I and I' lie on the internal angle bisector of <A.
I' lies also on the external angle bisectors of <B and <C. These
external angle bisectors are perpendicular to the respective
internal angle bisectors. So BI and BI' are perpendicular; CI and
CI' are perpendicular as well.
4. From step 3 we can conclude that triangles EBI and E'I'B are
similar. From this we can conclude from E'I'/E'B = EB/EI:
r' s-b
--- = ---
s-c r
and thus
r*r' = (s-b)(s-c) ....................[1]
5. We can also see that triangles AIG and AI'G' are similar. Here we
can conclude from IG/I'G' = AG/A'G' that:
r s-a
-- = --- .............................[2]
r' s
6. Multiplying [1] and [2] gives:
(s-a)(s-b)(s-c)
r^2 = ---------------
s
and
(s-a)(s-b)(s-c)
r = sqrt(---------------) ............[3]
s
7. It is not difficult to see that the area of ABC, let's call it K,
equals s*r. Combining this with [3] we find Heron's formula:
K = sqrt(s(s-a)(s-b)(s-c))
I hope this helped. If you need more help, just write us back.
Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/