The Plemelj Construction of a Triangle: 10
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
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 is an alternative to Plemelj's second construction. See The Plemelj Construction of a Triangle: 3.[more]
Draw a right-angled triangle with legs of length and , so that .
Step 1: Construct a circle with center through and, therefore, of radius . Construct the ray from such that the angle between and is . Let be the intersection of and .
Step 2: Let be the midpoint of , so that . Let be the midpoint of . Let be the intersection of the perpendicular bisector of and the line through parallel to .
Step 3: The triangle meets the stated conditions.
Since and is the perpendicular bisector of , and . So . The triangle is isosceles, so also and , and .
Since is isosceles, .
So and .[less]
Contributed by: Izidor Hafner (September 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.