The Plemelj Construction of a Triangle: 2

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Let be the midpoint of . By construction, , so .

By construction, , and the altitude from to has length .

It remains to prove .

Let the angle at be . Consider these three angles around the point : , and , which sum to . Since is parallel to and is the midpoint of , is the perpendicular bisector of , is isosceles, bisects and alternate angles around imply . Therefore . Since , . (1)

On the other hand, the chord subtends the angle from any point on above : . The quadrilateral is cyclic, so its opposite angles are supplementary: or . (2)

From (1) and (2), , so .


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



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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