The basic relation between points [x] of P

Relation a

First, fixing [x] and considering all [a]'s satisfying this relation.

Second, fixing [a] and considering all [x]'s satisfying this relation.

In the first case all [a]'s define a p-line in P

In the second case all [x]'s define a p-line of P

The important thing is that the set of lines of P

(1) line [a] = join([x],[y]), for points [x], [y] of P

(2) point [a] = inter([x]*,[y]*) , [x]*, [y]* being lines of P

Thus, in theorems of projective geometry, intechanging the words

A typical case is the theorem of Desargues (see Desargues.html ).

I and II below are equivalent.

(I) For triangles [x][y][z], [x'][y'][z'], lines [a]=join([x],[x']), [b]=join([y][y']), [c]=join([z][z']) are concurrent, i.e. there is a point [w] lying on all three lines [a], [b], [c].

(II) Intersection points of sides [p]=inter([xy],[x'y']), [q]=inter([yz],[y'z']), [r]=inter([zx],[z'x']) lie on the same line [e].

Here [xy] denotes the line [f]=join([x],[y]), [yz] denotes [g]=join([y],[z]), etc..

Assume we proved that I => II. Then to prove II => I take the dual of II:

[p]([xy],[x'y']) reads: line [p]* joining points [f] and [f'] (in the dual space P

[q]([yz],[y'z']) reads: line [q]* joining points [g] and [g'] (in the dual space P

[r]([zx],[z'x']) reads: line [r]* joining points [h] and [h'] (in the dual space P

lie on the same line [e] reads: [p]*,[q]*,[r]* intersect at [e] (again in P

Thus, II translates:

For triangles [f][g][h], [f'][g'][h'] of P

[u]([fg],[f'g']) etc. are on the same line [o]* (of P

Thus, the dual (II') of (II) is (I) reformulated in the projective space P

A similar situation arises in the theorem of Pappus, which is self-dual. i.e. the dual expresses the same figure as the original (see PappusSelfDual.html ). Another instance of duality is that between the theorems of Brianchon and Pascal.

Desargues.html

Menelaus.html

MenelausApp.html

PappusLines.html

PappusSelfDual.html

Pascal.html

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