Equilateral Triangle on Three Lines

What is this about?

17 July 2014, Created with GeoGebra

Problem

Given three straight lines (denoted below by two points $AB,$ $CD,$ $EF$).

Equilateral Triangle on Three Lines - problem

Construct an equilateral triangle with vertices one per line.

Construction

Choose an arbitrary point on one of the line, say $X$ on $EF.$ Rotate $CD$ around $X$ $60^{\circ}$ into $C'D'.$ Let $Y'$ be the intersection of $C'D'$ with $AB$ and $Y$ the point that was mapped into $Y'$ by the rotation.

Equilateral Triangle on Three Lines - solution

Then $\Delta XYY'$ is equilateral and $X\in EF,$ $Y'\in AB,$ and $Y\in CD.$

In general, the solution is not unique, e.g.,

Equilateral Triangle on Three Lines - extra example

Acknowledgment

The construction is classic. For three parallel lines (when the solution is unique up to isomorphism), it can be found in I. M. Yaglom, Geometric Transformations I (MAA, 1962), problem 18. The latter problem has been discussed elsewhere.

Related material
Read more...

Various Geometric Constructions

  • How to Construct Tangents from a Point to a Circle
  • How to Construct a Radical Axis
  • Constructions Related To An Inaccessible Point
  • Inscribing a regular pentagon in a circle - and proving it
  • The Many Ways to Construct a Triangle and additional triangle facts
  • Easy Construction of Bicentric Quadrilateral
  • Easy Construction of Bicentric Quadrilateral II
  • Star Construction of Shapes of Constant Width
  • Four Construction Problems
  • Geometric Construction with the Compass Alone
  • Construction of n-gon from the midpoints of its sides
  • Short Construction of the Geometric Mean
  • Construction of a Polygon from Rotations and their Centers
  • Squares Inscribed In a Triangle I
  • Construction of a Cyclic Quadrilateral
  • Circle of Apollonius
  • Six Circles with Concurrent Pairwise Radical Axes
  • Trisect Segment: 2 Circles, 4 Lines
  • Tangent to Circle in Three Steps
  • Regular Pentagon Construction by K. Knop
  • |Contact| |Front page| |Contents| |Geometry| |Store|

    Copyright © 1996-2017 Alexander Bogomolny

     62070393

    Search by google: