## 1. Parabola inscribed in a triangle

[1] A parabola (c) tangent to the sides of a triangle ABC has its perspector P on the outer Steiner ellipse of ABC.
[2] P is also on the inner Steiner ellipse of the cevian triangle A'B'C' of P with respect to ABC.
[3] The outer Steiner ellipse of ABC and the inner Steiner ellipse of A'B'C' are tangent at P. Their common tangent tP is line tr'(t(P)), where tr' denotes the trilinear polar with respect to A'B'C'. Point t(P) is the isotomic conjugate of P with respect to ABC and is a point at infinity. This point defines the direction of the axis of the parabola.

For [1] : Let L=tr(P) be the trilinear polar of P. For every point S on L the trilinear polar tr(S) with respect to ABC is a tangent to the parabola. Inversely the tripoles of tangents of (c) are on tr(P)=L. In our case the parabola has the line at infinity as a tangent, hence its tripole, which is the centroid G of ABC is on tr(P). Thus this line passes through the centroid, consequently its tripole P is on the outer Steiner ellipse of ABC.

For [2] : The conic with perspector P with respect to A'B'C' is generated by the tripoles Q = tr'(PS) of lines through P. If Q is on the line at infinity then its tripolar is tangent to the inner Steiner ellipse. Thus, ellipses, having no point at infinity have no line through P tangent to the in-ellipse. Analogously hyperbolas, having two points at infinity, have P outside the in-ellipse and parabolas, having one point at infinity have only one line through P tangent to the in-ellipse. Thus P is on the inner Steiner ellipse of A'B'C' and the tangent there is the trilinear polar tr'(P0) of the point at infinity P0 of the parabola.

For [3] : Is a consequence of the discussion in Autopolar2.html . From that discussion follows also that the axis of the parabola is in the direction GG', where G' is the centroid of A'B'C' and GG' is divided by tP in ratio 2:1.

The figure depicts some other important relations linked to a point P on the outer Steiner ellipse of a triangle ABC.
[1] Line L=tr(P)=tr(t(P0)) = ~P0 is the dual of P0 (point at infinity) isotomic of P. For every S on L the isotomic t(S) is on a hyperbola (c') circumscribing ABC with perspector P. This kind of hyperbolas is discussed in some detail in Trilinear_Polar.html . In particular tr(t(S)) passes through P0 hence is parallel to the axis of the parabola studied above. The hyperbola is the dual conic of the parabola, both conics having perspector with respect to ABC the point P (see TriangleConics.html ).
[2] By the fundamental commutativity of the operations tr*t = t*tr (see IsogonalGeneralized.html ), tr(t(S))=t(tr(S)) and since tr(S) is a tangent to the parabola we deduce the rule : the isotomic conjugates of the tangents of the inscribed to ABC parabola are parallel to its axis.
[3] For any point S on tr(P) points {tr'(SP), t'(S), P} are collinear lying on a line ~S' = tr'(t'(S')), where S' is another point on tr(P). ~S' is the polar of S' with respect to the parabola and the tangents to the parabola : tr(S) at t'(S) and the tangent at Q=tr'(SP) pass both through S' (see IsotomicGeneral.html ).
[4] In particular when S coincides with one of the intersection points {S1,S2} of the outer Steiner ellipse with tr(P) then point S' coincides with G, line ~S' coincides with tP, t(S) becomes a point at infinity of the hyperbola and SP becomes parallel to an asymptote of the hyperbola (see again Trilinear_Polar.html ).