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The Plemelj Construction of a Triangle: 14

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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.

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Construction

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.

Verification

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 .

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Contributed by: Gerd Baron, Izidor Hafner, Marko Razpet and Nada Razpet (August 2018)
Open content licensed under CC BY-NC-SA

Snapshots

Details

As far we know, this problem first appeared in [1]. The problem was also posed by Stevens in [2].

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.

References

[1] L. H. von Holleben and P. Gerwien, Aufgaben-Systeme und Sammlungen aus der Ebenen Geometrie: Aufgaben, Berlin: G. Reimer, 1832.

[2] The Ohio Journal of Education, (4), 1855 pp. 278 and 369; (5), 1856 p. 112; (6), 1857 pp. 56–57, 145 and 184.

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