The Rule of False Position and Geometric Problems - Construction of the Fourth Proportional

Let a, b, and c be three line segments.

On ray Or trace segments OA = a  and  AC = c. On ray Os trace segment OB = b. From point C trace CX parallel to AB. We can verify that x is the fourth proportional with respect to the segments a, b, and c. That is, a:b = c:x.

Indeed, triangles OAB and OCX  are similar. Therefore a:b = (a + c):(b + x).  So a:b = [(a + c) - a]:[(b + x) - b] = c:x, as desired.