x

- Through the tripols tr(L) of all lines L through a fixed point P

- Through the isotomic conjugates t(Q) of all points Q on the dual line ~P

The two ways are related: If Q

In fact the asserted collinearity is equivalent with the vanishing of the determinant with rows:

(1/x

(1/a, 1/b, 1/c) (point tr(L))

(1/(bz

Here ax+by+cz=0 denotes the variable line L through (x

Through an easy calculation the vanishing of this determinant reduces exactly to this last equation.

More relations between various lines and the conics related to to isogonal points {P

[1] For every point T on tr(P

[2] tr(P

[3] For every point T on tr(P

[4] Lines tr(T) and tr'(T) intersect at a point T' on tr(P

[5] The contact points T

[6] Analogously the tripolars of T', lines tr(T') and tr'(T') intersect at a point T''. Denote the contact points T

In IsogonalGeneralized.html it is discussed a generalization of the isotomy (as well as isogonality) based on the selection of an arbitrary point K instead of the centroid G defining the barycentric coordinates. For each such system the corresponding conjugation is defined and the remarks made above are also valid. One has only to interpret the resulting duality in the right context.

The direct proof using

- K has coordinates (1,1,1).

- tr(K) has equation x+y+z=0.

- a point P(x

Hence passes through (1,1,1).

[2] For P

a

as the image under t of the line a

IsogonalGeneralized.html

IsotomicChart.html

BarycentricCoordinates3.html

http://www.paideiaschool.org/TeacherPages/Steve_Sigur/resources/Kiepert%20and%20jerabek/Kiepert%20hyperbola.htm

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