The Plemelj Construction of a Triangle: 12
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 at . This construction is an alternative to The Plemelj Construction of a Triangle: 5.[more]
Draw a line segment of length and let the midpoint of be . Draw a line segment of length perpendicular to .
Step 1: Draw a circle with center so that the chord subtends the angle , which implies the central angle .
Step 2: Let be the intersection of the ray and the circle .
Step 3: Extend to so that is the midpoint of .
Step 4: Draw a ray parallel to at distance above . The point is the intersection of and .
Step 5: The triangle meets the stated conditions.
Let be the intersection of the segment and .
The quadrilateral is a parallelogram. The angle at is .
The triangle is isosceles, so the angle . On the other hand, . So and .[less]
For the history of this problem, references and a photograph of Plemelj's first solution, see The Plemelj Construction of a Triangle: 1.