In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.
Here is my sketch:
The triangle ##abc## is arbitrary, the triangles ##acp##, ##abq##, and ##bcr## are equilateral with centroids ##m##, ##n##, and ##k##. I suspect that the triangle ##mnk## is equilateral. Here is my proof.
By the equation for centroids,
##3m=a+c+p##
##3n=a+q+b##
##3k=b+c+r##...
##arg((b-a)/(c-a))## is an angle between ##ab## and ##ac##.
##arg((a-c)/(b-c))## is an angle between ##ca## and ##cb##.
For them to be equal, ##b## has to be equidistant from ##a## and ##c##, i.e. ##|b-a|=|b-c|##.
Then the equation for distances becomes, ##|b-a|/|c-a|=|c-a|/|b-a|##.
Thus...