[alogo] Isogonal of parabola - II

Given a triangle of reference ABC and a parabola (c) passing through its vertices, its isogonal conjugate is a tangent line tP of the circumcircle of ABC (see Remark-2 of IsogonalOfCircumcircle.html ) . Here I continue the discussion initiated in IsogonalOfParabola.html and investigate the parabola using the various facts on triangle conics referred to trilinear coordinates (see Trilinears.html and TriangleConics.html ).

[1] Let tP denote the tangent to the circumcircle at P, sP the Simson line of P, D the other intersection point with the circumcircle of the orthogonal to sP from P. D is on the parabola (c).
[2] Let QP be the tripole of line tP with respect to ABC and IP the isogonal conjugate of QP, cP the trilinear polar of IP. The sides of the triangle A'B'C' formed by the tangents to the parabola at the vertices of ABC intersect cP at points {A*,B*,C*} lying on the sides of ABC.
[3] The tripols AP of lines aP passing through IP are points of the parabola.
[4] The tripols BP with respect to A'B'C' of the tangents at AP are on the trilinear polar cP of IP.

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[1] was proved in IsogonalOfParabola.html .
[2] Is a general fact proved in IsogonalGeneralized.html .
[3,4] Are proved in TriangleConics.html .

Remark-1 Of particular interest is the construction of the tangential triangle A'B'C' in dependence of ABC and P. First can be constructed QP and then its isogonal conjugate IP. Then one can construct the trilinear polar cP of IP and determine its intersection points {A*,B*,C*} with the sides of ABC. Then one can construct the tangential triangle A'B'C' having sides the lines {AA*,BB*,CC*}.
Remark-2 From the tangential triangle A'B'C' and the knowledge of the axis direction PD of the parabola a standard procedure locates the focus and the directrix of the parabola using only line intersections.
For this draw parallels {kA,kB} from {A,B} to the axis direction PD and take their symmetrics {lA,lB} with respect to the tangents {AA*,BB*}. Later intersect at the focus F of the parabola. The directrix is found by taking the symmetric F' with respect to AA* and drawing the line passing through F' and orthogonal to the axis direction PD.
Remark-3 The focus and directrix of the parabola can be also constructed using the circumcircle of the tangential triangle A'B'C' and its orthocenter H. In fact H is a point on the directrix of the parabola and the circumcircle of A'B'C' passes through the focus F (see (4,5) of Miquel_Point.html ).
Remark-4 Taking the trilinear polar of an arbitrary point on line cP (the trilinear polar of IP) we have an additional fourth tangent of the parabola and the determination of the focus and directrix reduces to the Miquel circles of the aforementioned reference.
Remark-5 It can be shown that line PF passes through the centroid G of the tangential triangle A'B'C' which is contained in line cP. This can be also used to determine F as intersection of the circumcircle of A'B'C' and line PG.

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