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

Solution. Set the line segments and on a line so that is between and . Draw a half-circle with diameter (for finding the centre, see the entry midpoint). Let be the point where the normal line of passing through intersects the arc of the half-circle. The line segment 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 is a right triangle. Its height divides this triangle into two smaller right triangles which have equal angles with the triangle and thus are similar. Accordingly, we can write the proportion equation concerning the catheti of the smaller triangles

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