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The Plemelj Construction of a Triangle: 11

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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 at . This construction is an alternative to the one from Plemelj's teacher's textbook. See The Plemelj Construction of a Triangle: 2.

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Construction

Step 1: Draw a line segment of length . Draw the line segment of length perpendicular to . Let be the midpoint of .

Step 2: Construct a circle such that the chord subtends an angle from points on above the chord. Let be the center of . The corresponding central angle is .

Step 3: Let be the intersection of and the line through parallel to .

Step 4: The triangle meets the stated conditions.

Verification

By construction, .

Let be the midpoint of . By construction, , so and the altitude from to has length .

It remains to prove .

The angle , since and triangle is isosceles.

By construction, the chord subtends the angle from any point above the chord.

So .

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

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Details

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

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