construction of fourth proportional

Task.  Given three line segments $a$, $b$ and $c$.  Using compass and straightedge, construct the fourth proportional of the line segments.

Solution.  Draw an angle ($\alpha$) and denote its vertex (http://planetmath.org/Angle) by $P$.  Separate from one side (http://planetmath.org/Angle) of the angle the line segments  $PA=a$  and  $AB=b$, and from the other side of the angle the line segment  $PC=c$.  Draw the line $AC$ and another line parallel to it passing through $B$.  If the last line intersects the other side of the angle in the point $D$, then the line segment  $CD=x$  is the required fourth proportional:

$$a:b=c:x$$

Justification: the intercept theorem.

The below picture illustrates this solution:

Note.  The special case  $c=b$  gives the third proportional $x$ of $a$ and $b$:

$$a:b=b:x$$