The Plemelj Construction of a Triangle: 3
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 .[more]
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.
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 , .[less]
For the history of this problem, references and a photograph of Plemelj's first solution, see The Plemelj Construction of a Triangle: 1.