the complete quadrangle
Take four points in the plane, no three of them on a line, and draw every line you can through pairs of them. Four points pair up into six lines — and those six lines cross in three brand-new points besides the originals. This whole figure of four points, six lines, and three extra intersections is the complete quadrangle. It is the projective geometer's basic gadget for constructing special points with nothing but a straightedge.
Spell it out. Label the four points (the vertices) A, B, C, D, in general position. The six sides are the lines AB, CD, AC, BD, AD, BC. They naturally split into three opposite pairs: AB with CD, AC with BD, AD with BC. Each opposite pair meets in a point; those three meeting points P = AB . CD, Q = AC . BD, R = AD . BC are the diagonal points, and the triangle PQR they form is the diagonal triangle. The magic is that on each side of the diagonal triangle the figure marks out a perfect harmonic set: for instance, on the line through two diagonal points, the two vertices of the quadrangle lying on it are harmonic conjugates with respect to those diagonal points. This is precisely the straightedge-only recipe for the harmonic conjugate.
The complete quadrangle matters because it manufactures the harmonic relation (cross-ratio -1) using incidence alone — no measuring, no compass — which is why it is the foundation stone for projective constructions and the projective definition of conics. Note the careful naming: a complete quadrangle is FOUR points and their SIX connecting lines; its dual, the complete quadrilateral, is four LINES and their SIX intersection points — do not confuse the two, and do not confuse either with an ordinary four-sided quadrilateral, which has only four sides, not six. In the real projective plane the three diagonal points are never collinear (a fact that, in other 'characteristic 2' settings, can fail).
Vertices A, B, C, D form the quadrangle; its six sides give diagonal points P = AB . CD, Q = AC . BD, R = AD . BC. On line QR, the two vertices that lie on it together with Q and R make a harmonic set — so the construction hands you, with a ruler only, the harmonic conjugate of any chosen point with respect to two others.
Four points, six lines, three diagonal points — and a harmonic set on every side of the diagonal triangle.
Distinguish three figures: a complete quadrangle (4 points, 6 lines), its dual the complete quadrilateral (4 lines, 6 points), and an ordinary quadrilateral (4 points, 4 sides). The harmonic property of the diagonal triangle is the straightedge-only construction of harmonic conjugates and underlies the projective definition of a conic's pole and polar.