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20b. Construct a Triangle Given the Length of Its Base, the Difference of Its Base Angles and a Line Containing the Third Vertex

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This Demonstration constructs a triangle given the length of its base, the difference of the base angles and a line (given by a point and slope ) that contains . This generalizes the problem given to Plemelj when he was in high school, namely, the problem in which the line is parallel to . Our construction is a simple adaptation of The Plemelj Construction of a Triangle: 4.

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Construction

Step 1: Draw a line segment of length and its midpoint . Draw a line .

Step 2: Draw a line segment from that is perpendicular to . It meets the line at .

Step 3: Choose any point on the ray .

Step 4: Let be on such that .

Step 5: Let be the circle with center and radius .

Step 6: Let be the intersection of and the ray . Join and .

Step 7: The point is the intersection of and the line through parallel to .

Step 8: The triangle meets the stated conditions.

Verification

Theorem: Let be any triangle, and let be the foot of the altitude from to . Let be the circumcenter of , and let the segment be perpendicular to with on the same side of as . Then .

Proof: The inscribed angle subtended by the chord is , so the central angle . Since is isosceles, . Since , . Since , [1, Proposition 29]. So .

Now in the construction, let be the intersection of and the line through parallel to . Then the isosceles triangles and are similar, and is parallel to , so . By construction, , so . Since is on the right bisector of , it is the circumcenter of . By the theorem, .

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

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Details

This problem comes from [2], where some suggestions on how to solve it are also given. That it could be solved by Plemelj's construction was observed by Nada and Marko Razpet. For an algebraic solution, see 20. Construct a Triangle Given the Length of Its Base, the Difference of Its Base Angles and a Line Containing the Third Vertex.

For the history of Plemelj's problem, references and a photograph of Plemelj's first solution, see The Plemelj Construction of a Triangle: 1.

References

[1] Euclid, Elements, Vol. 1, 2nd ed. (T. Heath, ed.), New York: Dover Publications, 1956 pp. 311–312.

[2] E. Heis and T. J. Eschweiler, Lehrbuch der Geometrie zum Gebrauche an höheren Lehranstalten, Köln: Verlag der M. Dumont-Schauberg'schen Buchhandlung, 1855.

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