compass and straightedge construction of geometric mean

Given line segmentsMathworldPlanetmath of lengths a and b, one can construct a line segment of length ab using compass and straightedge as follows:

  1. 1.

    Draw a line segment of length a. Label its endpointsMathworldPlanetmath A and C.

  2. 2.

    Extend the line segment past C.

  3. 3.

    Mark off a line segment of length b such that one of its endpoints is C. Label its other endpoint as B.

  4. 4.

    Construct the perpendicular bisectorMathworldPlanetmath of AB¯ in order to find its midpointMathworldPlanetmathPlanetmathPlanetmath M.

  5. 5.

    Construct a semicircle with center M and radii AM¯ and BM¯.

  6. 6.

    Erect the perpendicularPlanetmathPlanetmath to AB¯ at C to find the point D where it intersects the semicircle. The line segment DC¯ is of the desired length.


    This construction is justified because, if AD¯ and BD¯ were drawn, then the two smaller trianglesMathworldPlanetmath would be similarMathworldPlanetmathPlanetmath, yielding


    Plugging in AC=a and BC=b gives that DC=ab as desired.

    If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.

Title compass and straightedge construction of geometric mean
Canonical name CompassAndStraightedgeConstructionOfGeometricMean
Date of creation 2013-03-22 17:14:55
Last modified on 2013-03-22 17:14:55
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Algorithm
Classification msc 51M15
Classification msc 51-00
Related topic ConstructionOfCentralProportion