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The Plemelj Construction of a Triangle: 6

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This Demonstration constructs a triangle given the length of its base, the length of the altitude from to and the difference between the angles at and .

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Construction

Step 1: Draw a vertical segment of length . Draw two horizontal rays and from and , respectively. From , draw a ray at angle with respect to the ray . Let be the intersection of the rays and .

Step 2: Draw a segment of length perpendicular to the ray .

Step 3: Draw a circle with center and radius . The ray intersects the circle at and .

Step 4: On , draw the point so that . On , measure out the point so that .

Step 5: The triangle satisfies the stated conditions.

Verification

Draw the circumcircle of the triangle with center . Let the point be the foot of the altitude from . Then . By construction, the power of the point with respect to is , so by Euclid III.37 [1, pp. 15], is tangent to and as angles with orthogonal legs. So .

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Contributed by: Izidor Hafner (August 2017)
Open content licensed under CC BY-NC-SA

Snapshots

Details

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

Reference

[1] G. E. Martin, Geometric Constructions, Springer, New York, 1998.

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