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# 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 $CD$ divides this triangle into two smaller right triangles which have equal angles with the triangle $ABC$ and thus are similar. 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 $q$.

## Mathematics Subject Classification

51M15*no label found*

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