Pompeïu's Theorem
Pompeïu's Theorem
This Demonstration gives a proof of Pompeïu's theorem: If is a point in the plane of the equilateral triangle , then there exists a triangle with side lengths , , and unless lies on the circumcircle of the triangle , when the triangle is degenerate.
D
ABC
AD
BD
CD
D
ABC
We rotate the equilateral triangle around the point by an angle of . This gives the triangle . As and , is equilateral, so . Also, and , so we have constructed whose sides have the same lengths as , , and .
ABC
C
π/3
A'B'C'
DC=D'C
∠DCD'=π/3
ΔDCD'
DD'=DC
AD=A'D'
BD=B'D'
ΔDBD'
AD
BD
CD