To make calculations shorter I use the following abridged notation :

Matrices C, C' are inverse to each other up to the factor -2.

Let for the moment P=diag(p

Introduce also the matrices:

C

C

C

C

Obiously C

In the rest I denote by P the triple (p

P

Lines are written in the form P

Conics are written in the form X

Adopting this notation the following expressions result:

The polar line of a conic at a point P has coefficients Q=P

[1] The tripolar tr(P) : (1/P)

[2] The isotomic transform t(P) of P : t(P) = 1/P .

[3] The tripolar of the isotomic or the

The four basic conics associated with ABC, P and t(P):

[4] The circmuconic with perspector P, c

[5] The circumconic with perspector t(P), c

[6] The inconic with perspector P, c

[7] The inconic with perspector t(P), c

[8] An arbitrary point Q on the tripolar tr(P) : Q

[9] Line PQ : A

[10] The intersection Q' of line PQ and ~P : Q' = P x (P x Q) .

[11] The tripole tr(PQ) of line PQ is on conic c

[12] The isotomic t(Q) of Q : t(Q) = 1/Q : (1/Q)

[13] The isotomic of Q' : t(Q') = 1/Q' : (1/Q')

[14] Collinearity (tr(PQ), t(Q'), P') on line L : (1/A x 1/Q')

[15] The isotomic t(PQ) of line PQ is tangent to the inconic c

[16] The tripolar tr(Q) of Q : 1/Q = D

[17] Collinearity (D,P,tr(PQ)) : D

[18] Intersection of the tangent at the tripole tr(PQ) of c

[19] Point S is the pole with respect to c

[20] Collinearity (H, t(Q), t(P)) on line t(tr(S)) : S

[21] Line C

[22] Intersection T of lines tr(Q) and tr(Q') : T = 1/Q x 1/Q' .

[23] Collinearity (F,T,~L) on tr(Q') : (1/Q')

[24] Collinearities (tr(L),t(Q),t(Q'),Q'') on ~T : T

[25] The tangent to c

[26] The polar of t(Q'') w.r to c

[27] Collinearity (S,D,T) on tr(Q) : (1/Q)

[28] Collinearity (~L,H,t(Q'')) on line t(PQ) : L

There are more relations (infinite many actually) hidden in the figure related to the sequence of triangles emerging from P and its harmonic associates and its repeated constructs (see TrilinearProjectivity.html ).

From the relations displayed most are trivial or/and proved elsewhere. The first statement to prove is [11] and is shown in (*) BarycentricCoordinates3.html . [12,13] are instances of the same property proved in IsogonalGeneralized.html . [14, 16] is proved in IsotomicConicOfLine.html . [15] follows trivially from the previous since t(PQ) tangent to inconic c

It is interesting to see how these relations specialize when Q is on the line PP'. This is examined in IsotomyChart.html .

IsogonalGeneralized.html

IsotomicConicOfLine.html

IsotomicGeneral.html

IsotomyChart.html

TrilinearProjectivity.html

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