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.