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The Plemelj Construction of a Triangle: 10

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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 is an alternative to Plemelj's second construction. See The Plemelj Construction of a Triangle: 3.

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Let .

Construction

Draw a right-angled triangle with legs of length and , so that .

Step 1: Construct a circle with center through and, therefore, of radius . Construct the ray from such that the angle between and is . Let be the intersection of and .

Step 2: Let be the midpoint of , so that . Let be the midpoint of . Let be the intersection of the perpendicular bisector of and the line through parallel to .

Step 3: The triangle meets the stated conditions.

Verification

Since and is the perpendicular bisector of , and . So . The triangle is isosceles, so also and , and .

Since is isosceles, .

So and .

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Contributed by: Izidor Hafner (September 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.

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