Trilinear Coordinates - Formulas

Formulas

Trilinears enable many algebraic methods in triangle geometry. For example, three points

P = p : q : r

U = u : v : w

X = x : y : z

are collinear if and only if the determinant

equals zero. The dual of this proposition is that the lines

pα + qβ + rγ = 0

uα + vβ + wγ = 0,

xα + yβ + zγ = 0

concur in a point if and only if D = 0.

Also, if the actual directed distances are used when evaluating determinant D, then (area of (PUX)) = KD, where K = abc/8σ2 if triangle PUX has the same orientation as triangle ABC, and K = - abc/8σ2 otherwise.

Many cubic curves are easily represented using trilinears. For example, the pivotal self-isoconjugate cubic Z(U,P), as the locus of a point X such that the P-isoconjugate of X is on the line UX is given by the determinant equation

Among named cubics Z(U,P) are the following:

Thomson cubic: Z(X(2),X(1)), where X(2) = centroid, X(1) = incenter

Feuerbach cubic: Z(X(5),X(1)), where X(5) = Feuerbach point

Darboux cubic: Z(X(20),X(1)), where X(20) = De Longchamps point

Neuberg cubic: Z(X(30),X(1)), where X(30) = Euler infinity point.