Here are the details regarding this classic construction from projective geometry:

Take a quadrangle (4 blue points in our case).

Make this into a `complete quadrangle' by drawing a line through every pair of points (so 6 blue lines are added).

The newly added lines meet at three extra points -the diagonal triangle of the quadrangle (so 3 yellow points, and three yellow lines are added).

Mark where edges of the diagonal triangle intersect with complete quadrangle edges (so 6 green points are added).

Draw all lines linking the six newly generated points (so 4 green lines are added).

The final four lines create a quadrilateral which is the dual of the initial quadrangle.
Also the initial quadrangle can be considered to be the dual of the quadrilateral.

1. Why Perspective Drawing Works
2. Without Equations, Conics & Spirals
3. Foundations & Tilings in Perspective
4. When Does A Parabola Look Like An Ellipse?
5. Desargues' Theorem Proof
6. Axioms, Duality and Projections
7. Conics Made Easily and Beautifully
8. Harmonic Quadrangles & The 13 Configuration
9. The Line Woven Net
10. Brianchon's Theorem (Pascal's Dual)
11. Five Points Define A Conic
12. Projective Transformations Of Lines
13. Involutions Of The Line
14. Constructing The Dual Of A Quadrangle - The Thirteen Point Configuration
15. Pascal's Hexagrammum Mysticum Theorem
16. Non Euclidean Geometry & Hyperbolic Social Networks

Protective geometry is deeper and more fundamental than standard euclidean geometry, and has many applications in fundamental physics, biology and perspective drawing. We shall introduce it visually, without relying upon equations. The hope is make this beautiful subject accessible to anybody, without requiring prior knowledge of mathematics. At the same time, there are some very deep, rarely discussed ideas in this subject which could also benefit experts.