circles superscribed and inscribed on a triangle

[Michael Robinson]

First:
You are given a triangle, with a circle drawn such that it intersects all
three vertices of the triangle. Call the lengths of the sides of the
triangle a, b and c and the circle's radius r. Then

r^2=(abc)^2/[(a^2+b^2+c^2)^2 - 2(a^4+b^4+c^4)]

Now, you have a circle inscribed within the same triangle such that the
circle just touches each side of the triangle at a single point. That is,
the triangle's sides are tangents to the circle. What equation relates r, a,
b and c?

[Robert Israel]

"Michael Robinson" <nos...@billburg.com> writes:

See e.g. <http://en.wikipedia.org/wiki/Incircle>
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

[Chip Eastham]

The radius of the inscribed circle is twice
the triangle's area divided by the triangle's
perimeter.

BTW, the circle that passes through the
vertices of a triangle is usually called
the circumscribed circle (terminology that
carries over to regular polygons, which
also have both circumscribed and inscribed
circles).

regards, chip