# 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