Drag any of the points A1, A2, A3
Three equilaterial triangles share a vertex. Their other vertices are connected as shown (in pink-colored lines). The mid-points of the pink-colored lines then form another equilateral triangle.
See: Matrin Gardner, The College Journal of Mathematics, 1999.
This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on July 15, 2002.