construction of central proportional

Task. Given two line segmentsMathworldPlanetmath p and q. Using compass and straightedge, construct the central proportional (the geometric mean) of the line segments.

Solution. Set the line segments  AD=p  and  DB=q  on a line so that D is between A and B. Draw a half-circle with diameterMathworldPlanetmathPlanetmath AB (for finding the centre, see the entry midpointMathworldPlanetmathPlanetmathPlanetmath). Let C be the point where the normal line of AB passing through D intersects the arc of the half-circle. The line segment CD is the required central proportional. Below is a picture that illustrates this solution:


(For more details on the procedure to create this picture, see compass and straightedge construction of geometric mean.)

Proof. By Thales’ theorem, the triangleMathworldPlanetmath ABC is a right triangle. Its height CD this triangle into two smaller right triangles which have equal angles with the triangle ABC and thus are similarMathworldPlanetmathPlanetmath ( Accordingly, we can write the proportion equation concerning the catheti of the smaller triangles


The equation shows that CD is the central proportional of p and q.

Note. The word catheti (in sing. cathetus) the two shorter sides of a right triangle.

Title construction of central proportional
Canonical name ConstructionOfCentralProportional
Date of creation 2013-03-22 17:34:14
Last modified on 2013-03-22 17:34:14
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Algorithm
Classification msc 51M15
Related topic GoldenRatio
Related topic CompassAndStraightedgeConstructionOfGeometricMean
Related topic ConstructionOfFourthProportional
Defines cathetus
Defines catheti