ParabolaInscribed

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 t_P 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 t_P in ratio 2:1.

2. Additional relations

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 t_P, 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 ).

1. Parabola inscribed in a triangle

2. Additional relations

See Also