MathDB

Each side of a triangle

ABC

is divided into three equal parts, and the middle segment in each of the sides is painted green. In the exterior of

\triangle ABC

three equilateral triangles are constructed, in such a way that the three green segments are sides of these triangles. Denote by

A',B',C'

the vertices of these new equilateral triangles that don’t belong to the edges of

\triangle ABC

, respectively. Let

A'',B'',C''

be the points symmetric to

A',B',C'

with respect to

BC,CA,AB

.
(a) Prove that

\triangle A'B'C'

and

\triangle A''B''C''

are equilateral. (b) Prove that

ABC,A'B'C'

, and

A''B''C''

have a common centroid.

Problem Statement