triangle problem ( alternative concept)

Total angle around Point P = 360 degrees
A, B and C are equidistant from point P (given in question)
if we rotate triangle ABC by 360/3 = 120 degrees, point A will move to current point B, Point B will move to current Point C and C will move to current Point A.
Another rotation by 120 degrees clockwise will take point A to Original Point C, point B to original Point A (Exactly what we are trying to find for this question) and so on. We have rotated twice by 120 degrees and hence a total of 240 degrees from the original position.
Only angle rotations that matter here are 120, 240 and 360 degrees because rotatting by any other degree will not align the 3 vertices to the original vertices.

Instead of a equilateral triangle, if it were a square and point P had been equidistant from the 4 vertices, the only angle rotations we should be bothered about are those at every 360 / 4 = 90 degrees.
so 90, 180, 270 and 360 would have been our concerned rotations.

Thank you!