IsotomyChart

Isotomy chart

In this figure I apply the results discussed in the file IsotomicChart.html (using the notation introduced there) to the case in which the variable line PQ, for Q on the tripolar tr(P), passes through the isotomic conjugate P'=t(P) of P. The proofs for these properties reduce to the corresponding more general properties in that file. Below I use the same enumeration to facilitate this correspondence. I start from property [8] since the first seven do not involve the particular position of line PQ.

[8] Line PP' : A^tX = 0, A = P x 1/P .
[9] Intersection point Q = tr(P)*PP' : Q = (1/P) x (P x 1/P).
[10] The intersection Q' of line PP' and ~P : Q' = P x (P x 1/P) .
[11] The tripole tr(PP') of line PP' : tr(PP') = 1/A , is an intersection point of conics c_0p and c_0p'.
[12] The dual point ~PP' : A = 1/P x P is the intersection of the tripolars : tr(P) and tr(P').
[13] The isotomic t(Q) of Q : t(Q) = 1/Q : (1/Q)^tC_0p'(1/Q) = 0 lies on conic c_0p'.
[14] The isotomic of Q' : t(Q') = 1/Q' : (1/Q')^tC_0p(1/Q') = 0 lies on conic c_0p.
[15] Collinearity (tr(PP'), t(Q'), P') : (1/A x 1/Q')^t(1/P) = 0, L = 1/Q' x 1/P .
[16] The isotomic t(PP') of line PP' : 1/A = H^tC_1p = H'^tC_1p' is tangent to both inconics c_1p and c_1p'.
[17] The tripolar tr(Q) of Q : 1/Q = D^tC_1p is tangent to c_1p.
[18] Collinearity (D,P,tr(PP')) : D^t(P x 1/A) = 0, C = P x 1/A .
[19] Intersection of the tangent at the tripole tr(PP') of c_0p and tr(Q) : S = ((1/A)^tC_0p) x 1/Q .
[20] Point S is the pole with respect to c_0p of line C^tX = 0 (see [18]). Thus C = S^tC_0p .
[21] Collinearity (H', 1/Q, 1/P) on line t(tr(S)) : S^tH' = S^t(1/Q) = S^t(1/P) =0.
[22] Line C^tX=0 (see [18]) is also the tripolar of a point Q'' on c_0p : Q'' = 1/C : Q''^tC_0pQ'' = 0.
[23] Collinearity on tr(Q') : (1/Q')^tF = (1/Q')^tT = (1/Q')^tP = (1/Q')^tL = (1/Q')^tR = 0.
[24] Intersection T of lines tr(Q) and tr(Q') : T = 1/Q x 1/Q' .
[25] Collinearities on ~T : T^tQ'' = T^t(1/Q') = T^t(1/Q) = T^t(1/L) = 0 .

Isotomy chart

See Also