First, we note that the yellow triangles are congruent. In fact, they all have a side with the same length and the angles adjacent to this side are equal since they are obtained by the trisection of the interior angles of the original regular polygon.
Then we note that the red triangles are also congruent. This happens since they all have an equal angle obtained by trisection of the interior angles of the regular polygon and the adjacent sides of this angle are equal, since the yellow triangles are congruent.
Therefore the sides of the green polygon are equal. Moreover, their interior angles are also equal since their amplitudes can be obtained by subtracting from 360º the measures of equal angles. Therefore, the green polygon is also regular, as the initial polygon.
In this proof we only use the fact that the initial polygon is regular and the division of its interior angles into three equal parts lead to equal angles. In fact, if we divide any of the interior angles of a regular polygon in three parts (not necessarily equal) and repeat the same division for all other angles, we still get a regular polygon with the same number of sides, as can be seen in the following applet: