Posted by: Dave Richeson | June 22, 2009

Carnot’s Theorem

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

Picture 1

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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Responses

  1. 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!

  2. Can I download the applet?

  3. can I ask again??he
    how to proof the theorem??

  4. [...] 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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