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The Plemelj Construction of a Triangle: 12

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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 Plemelj Construction of a Triangle: 5.

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Construction

Draw a line segment of length and let the midpoint of be . Draw a line segment of length perpendicular to .

Step 1: Draw a circle with center so that the chord subtends the angle , which implies the central angle .

Step 2: Let be the intersection of the ray and the circle .

Step 3: Extend to so that is the midpoint of .

Step 4: Draw a ray parallel to at distance above . The point is the intersection of and .

Step 5: The triangle meets the stated conditions.

Verification

Let be the intersection of the segment and .

The quadrilateral is a parallelogram. The angle at is .

The triangle is isosceles, so the angle . On the other hand, . 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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