construction of central proportional

Task. Given two line segments $p$ 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 diameter $AB$ (for finding the centre, see the entry midpoint). 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 triangle $ABC$ 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 similar (http://planetmath.org/SimilarityInGeometry). Accordingly, we can write the proportion equation concerning the catheti of the smaller triangles

$$p:CD=CD:q.$$

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