The Plemelj Construction of a Triangle: 14
The problem is to construct a triangle given the length of its base, the length of the altitude from to and the difference of the angles at and . This Demonstration shows Gerd Baron's solution of this construction problem using the Apollonius circle.[more]
Step 1: On line , choose a point . Draw a vertical segment of length . From , draw a ray at angle with respect to the segment . Let be the intersection of the ray and . From , draw a ray at angle and opposite to with respect to the segment . Let be the intersection of the ray and . The triangle is right angled. Let be the midpoint of .
Step 2: Draw a circumcircle of the triangle with center . This circle is the Apollonius circle of the triangle . Choose any point on the circle and draw a tangent of length with as the midpoint of .
Step 3: On , draw a point such that .
Step 4: On , draw the point so that . On , measure out the point so that .
Step 5: The triangle meets the stated conditions.
The correctness of the construction follows from properties of the Apollonius circle of the triangle . The radius of the Apollonius circle is and is independent of .[less]
As far we know, this problem first appeared in . The problem was also posed by Stevens in .
The photograph of Stevens's solution was taken from [2, Vol. VI, 1857 p. 56].
For the history of Plemelj's solutions of this problem, see The Plemelj Construction of a Triangle: 1.
 L. H. von Holleben and P. Gerwien, Aufgaben-Systeme und Sammlungen aus der Ebenen Geometrie: Aufgaben, Berlin: G. Reimer, 1832.
 The Ohio Journal of Education, (4), 1855 pp. 278 and 369; (5), 1856 p. 112; (6), 1857 pp. 56–57, 145 and 184.