Incidence (geometry) - Concurrence

Concurrence

Three lines in a projective plane are concurrent if all three of them intersect at one point. That is, given lines L₁, L₂, and L₃; these are concurrent if and only if

If the lines are represented using homogeneous coordinates in the form _L with m being slope and b being the y-intercept, then concurrency can be restated as

Theorem. Three lines L₁, L₂, and L₃ in a projective plane and expressed in homogeneous coordinates are concurrent if and only if their scalar triple product is zero, viz. if and only if

Proof. Letting g denote the duality mapping, then

The three lines are concurrent if and only if

According to the previous section, the intersection of the first two lines is a subset of the third line if and only if

Substituting equation (1) into equation (2) yields

but g distributes with respect to the cross product, so that

and g can be shown to be isomorphic w.r.t. the dot product, like so:

so that equation (3) simplifies to

Q.E.D.