construction of central proportional
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 this triangle into two smaller right triangles which have equal angles with the triangle and thus are similar (http://planetmath.org/SimilarityInGeometry). Accordingly, we can write the proportion equation concerning the catheti of the smaller triangles
The equation shows that is the central proportional of and .
Note. The word catheti (in sing. cathetus) the two shorter sides of a right triangle.
|Title||construction of central proportional|
|Date of creation||2013-03-22 17:34:14|
|Last modified on||2013-03-22 17:34:14|
|Last modified by||pahio (2872)|