Here’s a neat theorem from geometry.
Begin with any triangle. Let R be the radius of its circumscribed circle and r be the radius of its inscribed circle. Let a, b, and c be the signed distances from the center of the circumscribed circle to the three sides. The sign of a, b, and c is negative if the segment joining the circumcenter to the side does not pass through the interior of the triangle (such as the value b shown below, represented by the teal segment), and it is positive otherwise.
Then we have the following elegant result:
Carnot’s theorem. a+b+c=R+r
Check out this GeoGebra applet that I created to see this theorem in action.
Recently I wrote about the Japanese Theorem. If you were unsuccessful in proving this beautiful theorem, try again using Carnot’s Theorem.
Pretty good post. I just stumbled upon your blog and wanted to say
that I have really liked reading your posts. In any case
I’ll be subscribing to your feed and I hope you write again soon!
By: Maria on June 23, 2009
at 11:11 pm
Can I download the applet?
By: Heru on February 10, 2010
at 9:41 pm
Sure, no problem. The URL for the GeoGebra file is http://users.dickinson.edu/~richesod/carnot/carnot.ggb
By: Dave Richeson on February 10, 2010
at 10:10 pm
thanks ^^
By: Heru on April 15, 2010
at 7:46 pm
can I ask again??he
how to proof the theorem??
By: Heru on April 15, 2010
at 7:56 pm
[...] wrote a blog posts about two beautiful theorems from geometry: the so-called Japanese theorem and Carnot’s theorem. Today I finished a draft of a web article that looks at both of these theorems in more detail. It [...]
By: Japanese theorem for nonconvex polygons « Division by Zero on June 22, 2011
at 2:18 pm