General Constructions

Solutions for the three general challenges can be found at the links below.

General 1
General 2
General 3
General 4

The challenges in this section begin only with points on curve and tangent lines. Each solution may be an ellipse or a hyperbola, depending on the positions of the given objects. A complete constuction should be able to shift from on form to the other by dragging the given objects. There are no parabolas here, because those would only arise from special, limiting conditions.

Here I also want to go over a few rather advanced constructions that apply to all of the challenges in this section. Any five points in the plane, no three of them collinear, define a unique conic section. This curve can be constructed in Sketchpad with the use of Pascal’s Theorem, and I used to do so often. It later came to my attention that Isaac Newton had a very different construction, and I now prefer it. (Sorry, Blaise.) Newton also had a construction for the tangent line through a given point on the conic section. Both of these constructions were published in Principia. The reason for the tangent construction is that it will be needed for construction of certain parameters (vertices, foci, etc.). After consideration of the proofs, I have decided instead to cite references for the proofs and show only the constructions here.

Newton’s 5-Point and Tangent Line Constructions

Conic Parameter Constructions

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