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The Plemelj Construction of a Triangle: 3

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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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Let .

Construction

Step 1: Draw a right-angled triangle with legs of length and , so that .

Step 2: Construct a circle with center through (therefore of radius ). Construct the ray from such that the angle between and is . Let be the intersection of and .

Step 3: Let be the midpoint of , so . Let be the midpoint of and let be the intersection of , the right bisector of , with the line through parallel to .

Step 4: The triangle meets the stated conditions.

Verification

Since , and . The line is the right bisector of , so . So .

Now , so . is isosceles, so also. Then . Therefore in the isosceles triangle , the equal angles at and are . On the other hand, from the right angle at , . 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.

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