Pompeïu's Theorem

This Demonstration gives a proof of Pompeïu's theorem: If

D

is a point in the plane of the equilateral triangle

ABC

, then there exists a triangle with side lengths

AD

,

BD

, and

CD

unless

D

lies on the circumcircle of the triangle

ABC

, when the triangle is degenerate.

We rotate the equilateral triangle

ABC

around the point

C

by an angle of

π/3

. This gives the triangle

A'B'C'

. As

DC=D'C

and

∠DCD'=π/3

,

ΔDCD'

is equilateral, so

DD'=DC

. Also,

AD=A'D'

and

BD=B'D'

, so we have constructed

ΔDBD'

whose sides have the same lengths as

AD

,

BD

, and

CD

.