Three rounds on a pentagon

Each round raises an isosceles triangle with a fixed apex angle on every side of the current pentagon, and joins the five apexes into a new pentagon for the next round. Each round has its own colour: blue, then coral, then gold. By default the apex angles are 72°, 144° and 216°, the last being an apex of 144° raised inward. Drag Rounds done from 0 to 3, or press play: the pentagons from the first two rounds are generally not regular, and then the third round makes a regular pentagon at once, with the longest over the shortest at 1.000000 in the corner and every interior angle 108°. Drag the five vertices into a pentagon as lopsided as you like and the third round still comes out regular; only a few special shapes shrink to a single point, such as five evenly spaced points on a circle joined in the order of a star. Change the order of the rounds and the result is the same. This is the theorem of Petr, Douglas and Neumann: on a polygon with n sides, n − 2 rounds with apex angles of 1/n, 2/n, … up to (n − 2)/n of 360°, each used once. On a triangle that is a single round of 120°, whose apexes are the centres of the equilateral triangles on the sides: Napoleon's theorem.