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.

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Posted in Math | Tags: applet, Carnot's theorem, circle, circumcenter, circumscribed, GeoGebra, geometry, incenter, inscribed, Japanese theorem, triangle

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:

Mariaon June 23, 2009at 11:11 pm

Can I download the applet?

By:

Heruon February 10, 2010at 9:41 pm

Sure, no problem. The URL for the GeoGebra file is http://users.dickinson.edu/~richesod/carnot/carnot.ggb

By:

Dave Richesonon February 10, 2010at 10:10 pm

thanks ^^

By:

Heruon April 15, 2010at 7:46 pm

can I ask again??he

how to proof the theorem??

By:

Heruon April 15, 2010at 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 [...]

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Japanese theorem for nonconvex polygons « Division by Zeroon June 22, 2011at 2:18 pm