 # The Plemelj Construction of a Triangle: 2

Initializing live version Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Construction

[more]

Verification

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]

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

## Snapshots   ## Details

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

## Permanent Citation

Izidor Hafner

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send