IsotomicChart

Isotomic chart

Here is a figure displaying some of the basic constructs related to a triangle ABC, its isotomic transformation (t), the trilinear polar/pole operation (tr) and an arbitrary point P(p₁,p₂,p₃). Barycentric coordinates are used throughout this review. G denotes the centroid of ABC. Starting with an arbitrary point Q on tr(P) the quantity of coincidence relations emerging is overwhelming. Only a few of these relations are listed below.
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(p₁,p₂,p₃), 1/P=diag(1/p₁,1/p₂, 1/p₃) denote a diagonal matrices.
Introduce also the matrices:
C_0p = PCP,
C_0p' = (1/P)C(1/P),
C_1p = (1/P)C'(1/P),
C_1p' = PC'P.
Obiously C_0pC_1p = C_0p'C_1p' = -2I (inverses up to factor -2).
In the rest I denote by P the triple (p₁,p₂,p₃) and by 1/P the triple (1/p₁,1/p₂,1/p₃).
P^tQ denotes the usual inner product (p₁q₁+p₂q₂+p₃q₃) and P x Q the vector product. Thus, S=P x Q means s₁=p₂q₃-p₃q₂, ... etc.
Lines are written in the form P^tX = 0 and their coefficients identified with the tripple P.
Conics are written in the form X^tMX=0, M being the matrix of the conic.
Adopting this notation the following expressions result:

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

[1] The tripolar tr(P) : (1/P)^tX = 0.
[2] The isotomic transform t(P) of P : t(P) = 1/P .
[3] The tripolar of the isotomic or the dual of P, ~P = tr(t(P)) : P^tX = 0 .

The four basic conics associated with ABC, P and t(P):
[4] The circmuconic with perspector P, c_0p : X^tC_0pX = 0.
[5] The circumconic with perspector t(P), c_0p' : X^tC_0p'X = 0.
[6] The inconic with perspector P, c_1p : X^tC_1pX = 0.
[7] The inconic with perspector t(P), c_1p' : X^tC_1p'X = 0 .

[8] An arbitrary point Q on the tripolar tr(P) : Q^t(1/P)=0.
[9] Line PQ : A^tX = 0, A = P x Q .
[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_0p : tr(PQ) = 1/A , (1/A)^tC_0p(1/A) = 0 .
[12] The isotomic t(Q) of Q : t(Q) = 1/Q : (1/Q)^tC_0p'(1/Q) = 0 lies on conic c_0p'.
[13] The isotomic of Q' : t(Q') = 1/Q' : (1/Q')^tC_0p(1/Q') = 0 lies on conic c_0p.
[14] Collinearity (tr(PQ), t(Q'), P') on line L : (1/A x 1/Q')^t(1/P) = 0, L = 1/Q' x 1/P .
[15] The isotomic t(PQ) of line PQ is tangent to the inconic c_1p' : 1/A = H^tC_1p' .
[16] The tripolar tr(Q) of Q : 1/Q = D^tC_1p is tangent to c_1p.
[17] Collinearity (D,P,tr(PQ)) : D^t(P x 1/A) = 0, C = P x 1/A .
[18] Intersection of the tangent at the tripole tr(PQ) of c_0p and tr(Q) : S = ((1/A)^tC_0p) x 1/Q .
[19] Point S is the pole with respect to c_0p of line C^tX = 0 (see [17]). Thus C = S^tC_0p .
[20] Collinearity (H, t(Q), t(P)) on line t(tr(S)) : S^tH = S^t(1/Q) = S^t(1/P) =0.
[21] Line C^tX=0 (see [17]) is also the tripolar of a point Q'' on c_0p : Q'' = 1/C : Q''^tC_0pQ'' = 0.
[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')^tF = (1/Q')^tT = (1/Q')^tL = 0.
[24] Collinearities (tr(L),t(Q),t(Q'),Q'') on ~T : T^tQ'' = T^t(1/Q') = T^t(1/Q) = T^t(1/L) = 0 .
[25] The tangent to c_0p' at t(Q) passes through t(Q'') : (1/Q)^tC_0p'(1/Q'')=0 .
[26] The polar of t(Q'') w.r to c_0p' passes through t(Q) and P' : (1/Q'')^tC_0p' = 1/Q x 1/P.
[27] Collinearity (S,D,T) on tr(Q) : (1/Q)^tS = (1/Q)^tD = (1/Q)^tT = 0.
[28] Collinearity (~L,H,t(Q'')) on line t(PQ) : L^t(1/A)=H^t(1/A)=(1/Q'')^t(1/A) =0.

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_1p' is equivalent with tr(t(PQ)) = ~PQ is a point on tr(t(P)) : P^tX=0, i.e. P^tA=0 which is true. Using the property of tr(P) to be common polar of c_0p and c_1p and the fact that every point Q on it has the same polar with respect to the last two conics, one can easily show the collinearity of (P,D,tr(PQ)) ([17]) and analogously of (P',H,t(Q)) [26]. From these all other relations follow by easy calculations.
It is interesting to see how these relations specialize when Q is on the line PP'. This is examined in IsotomyChart.html .

Isotomic chart

See Also