The Plemelj Construction of a Triangle: 2
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 .[less]
For the history of this problem, references and a photograph of Plemelj's first solution, see The Plemelj Construction of a Triangle: 1.