# Algebraic Solution of the Plemelj Triangle Construction Problem

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This problem asks for the construction of a triangle with ruler and compass given the length of the base (for simplicity, let ), the length of the altitude from to and the difference of the angles and . The problem was posed to Plemelj by his math teacher, Borstner, when Plemelj was about 16 years old.

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Contributed by: Izidor Hafner (June 2017)

Open content licensed under CC BY-NC-SA

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## Details

Plemelj's most original contribution is the elementary solution he provided for the Riemann–Hilbert problem about the existence of a differential equation with given monodromy group [1, 2]. Plemelj's first solution of the triangle construction problem was given in [2]. We take his trigonometry explanation from [3, pp. 191]. He also noted that he had made nine original solutions of the problem, and that he had also known about two textbook solutions.

The text in [3] is from a talk Plemelj gave in 1949 at the First Congress of Yugoslav Societies of Mathematicians, Physicists and Astronomers in Bled (Slovenia). The talk was published in 1951 in Belgrade in the proceedings of the congress.

Two different solutions of the problem are published in [4], and one in [5].

References

[1] Wikipedia. "Josip Plemelj." (May 31, 2017) en.wikipedia.org/wiki/Josip_Plemelj.

[2] J. J. O'Connor and E. F. Robertson. "Josip Plemelj." MacTutor. www-history.mcs.st-andrews.ac.uk/Biographies/Plemelj.html.

[3] J. Plemelj, *Iz mojega življenja in dela* (*From My Life and Work*), *Obzornik mat. fiz*., 39, 1992 pp. 188–192.

[4] D. S. Modic, *Triangles, Constructions, Algebraic Solutions* (in Slovenian), Ljubljana: Math Publishers, 2009.

[5] I. Pucelj, "Plemelj's Triangle and Fixed Points of Transformations" (in Slovenian), *Obzornik mat. fiz.*, 62(1), 2015 pp. 12–14.

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