Lazare Nicolas Marguérite Carnot (1753-1823) was a person of many gifts and strong political views. During the Revolution he directed the Army of the North and under Napoleon Bonaparte he served as the Minister of War. In 1794, together with Gaspard Monge he founded the now famous Ecole Polytechnique. His geometric and military publications are considered masterpieces in their respective areas.

The nice little theorem that bears his name shows an unexpected relationship between the radii of the circumcircle and incircle of a given triangle.

In the formulation and the proof of Carnot's theorem I'll use the standard notations. The theorem read as follows

In acute triangles, the circumcenter is always located inside the triangle. In this case, all three segments OM_a, OM_b, and OM_c lie entirely inside the triangle. If one of the angles is obtuse, the circumcenter falls outside the triangle. One of the segments (the one that corresponds to the side opposite the obtuse angle) lies entirely outside, while the other two segments lie only partially outside the triangle. In the above sum, the segments that intersect the interior of the triangle are taken with the sign plus, the remaining side with the sign minus. Segments OM_a, OM_b, and OM_c serve as altitudes of triangles OBC, OAC, and OAB with the bases on the sides a, b, and c. The sign convention guarantees that the areas of these triangles (taken with the proper signs) always add up to the area of ABC.

Proof

I shall consider only the case of an acute triangle. The other case is left as an exercise.

For the inradius we always have

From the above, we also have Area(OAB) + Area(OBC) + Area(OAC) = Area(ABC). Therefore,

A remark is now in order. O being the circumcenter of ABC, in the isosceles triangle AOB, AOB = 2C. And similarly, BOC = 2A and AOC = 2B. Equipped with this knowledge we may consider several triples of similar (right-angled) triangles:

From the first triple we derive

which leads to

Similarly, we get two additional identities

Summing the three up yields after simplifications

Adding this to (*) and dividing by (a + b + c) completes the proof.

References

1. R. Honsberger, Mathematical Gems III, MAA, 1985

Copyright © 1996-2009 Alexander Bogomolny