THE LABORATORY / GEOMETRY
A theorem is more interesting
when you can move it.
Drag six points. Follow three intersections.
See what stays true.
A live mathematical construction
EXPERIMENT 001Drag a letter · Arrow keys move a focused point
For six distinct points on a nondegenerate conic, the three intersections of opposite sides of the resulting hexagon are collinear in the projective plane. The labels need not run around the ellipse in order.
This lab uses an ellipse. An ellipse, a parabola, and a hyperbola are all nondegenerate conics. Projective geometry relates these different affine shapes.
Projective geometry adds points at infinity. Parallel lines share a direction, and meet at the corresponding point at infinity. In the symmetric preset, all three intersections lie at infinity: their Pascal line is the line at infinity, which cannot be drawn as an ordinary finite line.
Coincident points can make this construction undefined. The lab reports these configurations rather than inventing a line.
Each point is represented in homogeneous coordinates. Cross products compute every side and intersection. A reference line is formed from two intersections; the third is tested independently.
Read a projective proof
Explore a university proof worksheet
Predict. Test. Explain.
YOUR NEXT MOVE01 / MAKE A PREDICTION
Keep all six points distinct and on the ellipse, with the construction defined. Imagine dragging C to a new position.
Moving C along the ellipse keeps all six points on one conic. Pascal’s theorem preserves collinearity; it does not fix the line’s location. A finite experiment illustrates the claim. A proof explains why it holds.
Return to the lab and test your prediction02 / TRY A FRESH SETTING
Make a prediction on the left to open a new setting.
Six distinct points lie on a parabola. Opposite sides meet at three distinct finite points. Must those intersections be collinear?
An ellipse and a parabola are both nondegenerate conics. The theorem’s condition applies to either curve. The distinct, defined intersections in this question give one common line.
In your own words: which condition stayed the same, and which visible feature changed?
AI as a workbench. Use it to explore questions, examples, and constructions. Keep your reasoning in charge: predict, test, explain. A convincing animation is a starting point for a proof.
Small experiments. Serious ideas.
THE WORKBENCHFit a regular polygon inside a unit circle.
What happens to its area as the sides increase?
An = n/2 sin(2π/n) → π
Bring two points on y = x² together.
Watch a secant approach the tangent.
[f(0.6 + h) − f(0.6)] / h → 1.2
Less watching.
More finding out.