Answer to the first challenge by Anthony OR - GGG 2015

Problem: Given are three parallel lines and . Construct an equilateral triangle such that each of its vertices is on one of the lines. Solution: For any equilateral triangle a rotation with center one of the vertices and angle will map one of the remaining vertices onto the third vertex. Let be an arbitrary point on line .
  • Drag the slider to the end to rotate line on angle around point .
If vertex is on line , then the third vertex of the triangle should be on the image of line , but it also has to be on the third line . Therefore, vertex is the intersection point of the lines and . This way we find two vertices and determine the side of the equilateral triangle.
  • Click on the Construction button to finish the construction.
There will be two different solutions corresponding to the two possible directions of the rotation, at and degrees.
  • Drag the gray points to change the positions of the lines.
  • What do you notice when line is above line or below line ?
A geometric construction using this transformation was first described by I. M. Yaglom, in Geometric Transformations I, MAA, 1962, Chapter 2, Problem 18