Carnot's theorem (perpendiculars)

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

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

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.

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

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 ${\displaystyle |AP_{c}|=|BP_{c}|}$, ${\displaystyle |BP_{a}|=|CP_{a}|}$ and ${\displaystyle |CP_{b}|=|AP_{b}|}$ and therefore the equation above holds. which means all three perpendicular bisectors intersect in the same point.

• 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.

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