The interactive mechanical linkage below appears in the work of Frans van Schooten, a Dutch mathematician who lived in the 17th century.
Rod DP is attached to a board at point C and pivots around this point. Point F is also attached to the board and remains stationary. The linkage was constructed with the following specifications: DG = CF and CD = GF.
As you drag point D, notice that point P traces what looks to be a piece of a hyperbola branch.
Press the 'Show' button above to reveal segments CG and CF. Can you prove that triangle CDG is congruent to triangle GFC?
Now prove that triangle PDG is congruent to triangle PFC.
Why is PC = PG?
Use what you've proved to explain why point P traces a hyperbola branch. Hint: can you show that PF - PC is constant?
Compare this linkage to the Congruent Triangles construction for an ellipse.