Carnot's theorem (perpendiculars)

For other uses, see Carnot's theorem (disambiguation).

Carnot's theorem (named after Lazare Carnot) describes a necessary and sufficient condition for three lines that are perpendicular to the (extended) sides of a triangle having a common point of intersection. The theorem can also be thought of as a generalization of the Pythagorean theorem.

Carnot's theorem: if three perpendiculars on triangle sides intersect in a common point F, then
blue area = red area

Theorem

For a triangle $\triangle ABC$ with sides $a,b,c$ consider three lines that are perpendicular to the triangle sides and intersect in a common point $F$. If $P_{a},P_{b},P_{c}$ are the pedal points of those three perpendiculars on the sides $a,b,c$, then the following equation holds:

$|AP_{c}|^{2}+|BP_{a}|^{2}+|CP_{b}|^{2}=|BP_{c}|^{2}+|CP_{a}|^{2}+|AP_{b}|^{2}$

The converse of the statement above is true as well, that is if the equation holds for the pedal points of three perpendiculars on the three triangle sides then they intersect in a common point. Therefore, the equation provides a necessary and sufficient condition.

Special cases

If the triangle $\triangle ABC$ has a right angle in $C$, then we can construct three perpendiculars on the sides that intersect in $F=A$: the side $b$, the line perpendicular to $b$ and passing through $A$, and the line perpendicular to $c$ and passing through $A$. Then we have $P_{a}=C$, $P_{b}=A$ and $P_{c}=A$ and thus $|AP_{b}|=0$, $|AP_{c}|=0$, $|CP_{a}|=0$, $|CP_{b}|=b$, $|BP_{a}|=a$ and $|BP_{c}|=c$. The equation of Carnot's Theorem then yields the Pythagorean theorem $a^{2}+b^{2}=c^{2}$.

Another corollary is the property of perpendicular bisectors of a triangle to intersect in a common point. In the case of perpendicular bisectors you have $|AP_{c}|=|BP_{c}|$, $|BP_{a}|=|CP_{a}|$ and $|CP_{b}|=|AP_{b}|$ and therefore the equation above holds. which means all three perpendicular bisectors intersect in the same point.

References

• Wohlgemuth, Martin., ed. (2010). Mathematisch für fortgeschrittene Anfänger : Weitere beliebte Beiträge von Matroids Matheplanet (in German). Heidelberg: Spektrum Akademischer Verlag. pp. 273–276. ISBN 9783827426079. OCLC 699828882.
• Alfred S. Posamentier; Charles T. Salkind (1996). Challenging Problems in Geometry. New York: Dover. pp. 85–86. ISBN 9780486134864. OCLC 829151719.

External links

• Florian Modler: Vergessene Sätze am Dreieck - Der Satz von Carnot at matheplanet.com (German)
• Carnot's theorem at cut-the-knot.org
• Carnot's theorem at Interactive Geometry