Given a triangle of reference ABC and a parabola passing through its vertices, its isogonal conjugate is a tangent line of the circumcircle of ABC. Here I start inversely from a tangent to the circumcircle at P and discuss the way to recover the parabola from it. Let P be a point on the circumcircle and (c) be the isogonal conjugate parabola of the tangent tP at P. Then from the discussion in IsogonalOfCircumcircle.html it is known that
[1] the axis of the parabola is orthogonal to the Simson line of P.
Here I show that
[2] the fourth intersection point D of the parabola and the circumcircle is the intersection of the circle with the line parallel to the axis passing through P.
In fact, by the property of the circumcircle the point at infinity of tP will map under conjugation to a point D such that the Simson line of D is orthogonal to tP. Thus, D will be diametral to D' whose Simson line is parallel to tP. Thus, to find D draw a parallel from A to tP intersecting the circumcircle at A'. Then draw an orthogonal to BC intersecting the circle at D' and take the diametral D of D' (see SimsonProperty.html and SimsonDiametral.html ).
To see that PD is orthogonal to P1P2 or equivalently to AP' notice that OP is orthogonal to AA' and angle(OPD)=angle(ODP)=angle(A'AP'). Thus AP' is also orthogonal to PD.
The figure displays an additional property of the parabola:
[3] It passes through points like E, which are the intersections of parallels to the axis of the parabola drawn from the vertices of the anticomplementary triangle A1B1C1 with the opposite sides of the anticomplementary (see AnticomplementaryAndCircumparabola.html ).
Remark-1 Having that many points on the parabola one can draw it easily using means other than the isogonal transformation of the line. For example one could use Pascal's theorem and determine the tangents at the vertices of ABC, from them locate the focus and the directrix of the parabola etc.. There is though a simpler method using the general facts on triangle conics as these are exposed in TriangleConics.html . The corresponding discussion is to be found in IsogonalOfParabola2.html . Remark-2 The parabola is completely determined by its fourth intersection point D with the circumcircle. In fact, D has corresponding Simson line orthogonal to the tangent and this can be used to locate the tangent tP and from this point P, the axis and the whole parabola.
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