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The Plemelj Construction of a Triangle: 13

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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 and at and . This is not one of Plemelj's original constructions, but a new one based on his equation , where and . It is the same as The Plemelj Construction of a Triangle: 8, but the verification is different.

[more]

The modified equation is .

Construction

Step 1: Draw a straight line of length and a line parallel to at distance . Let be the midpoint of , and let be the point on directly above . Let be the reflection of in .

Step 2: Draw a circle with center and radius .

Step 3: Draw the ray from at the angle from to intersect at the point .

Step 4: Draw the circle with center and radius .

Step 5: Measure out a point on the circle at distance from .

Step 6: The point is the intersection of and the right bisector of .

Step 7: The triangle meets the stated conditions.

Verification

Angle , so . But , since .

So . Since , .

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Contributed by: Izidor Hafner, Nada Razpet and Marko Razpet (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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