Answer to the first challenge by Anthony OR - GGG 2015

Author:: Irina Boyadzhiev

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

Answer to the first challenge by Anthony OR - GGG 2015

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