Euclidean Geometry

Equilateral triangle and square have their origin in two circles of equal size that have been shifted along a straight line so that the distance between their centers equals their radius.
Equilateral triangles: The intersections of the two circles with the line form the vertices of two equilateral triangles (black triangles, center). These are created by connecting the intersection points with the centers of the circles.
Square construction: If the two intersection points of the circles are connected, a line is created that serves as the diameter of a smaller circle (shaded circle). The intersections of this smaller circle with the two larger circles and with the line define the vertices of a square (gray shaded, center).
Length ratios: The line connecting the intersections of the large circles (gray vertical, center) corresponds to twice the sine of 60°. This line divides the distance between the centers of the circles, which equals the radius, into two equal parts corresponding to the cosine of 60°. This division occurs at one quarter and three quarters of the diameter along the center line.
Construction of a hexagon and dodecagon: From the intersections of the two circles a line can be drawn to their intersection with the straight line. An isosceles triangle is formed whose vertex angle is 120°. Where this triangle intersects the inner circle, a regular hexagon is created (black).
By extending the construction with additional circles whose centers are each shifted by one radius length along the line (mouseover), the intersections of the four circles can be used to draw parallel lines. This construction produces a regular dodecagon. The intersections of the two middle circles form the center of these dodecagons (black stars).
With this method, using only two circles and optionally two additional auxiliary circles, regular triangles, squares, hexagons, and dodecagons can be constructed. The necessary circles create a total of seven relevant intersection points on the line.