By well known properties of parabolas, tangent to triangles, (look Miquel_Point.html ), the focus F of the parabola will lie on the circumcircle. F will be the common intersection point of the four circumcircles of the triangles ABC, IDC, HAI and BDH. In addition, all the orthocenters of these triangles will lie on the directrix of the parabola. This identifies the directrix with DE (passes through the orthocenter of ABC). These remarks suffice for the determination of the parabola.

The remarks that follow are related to the figure studied in the file MedialLine.html . F is symmetric to the common point E of the circumcircle of ABC with the line EJ joining the middle J of HI with A. E lies on the directrix DG and JE is equal to JF and orthogonal to the directrix, thus a point of the parabola. This identifies J as the point of contact with HD and AJ as parallel to the axis of symmetry of the parabola.

The parabola studied here has relations to the problem discussed in Olympiad1.html . There we consider segments joining points of sides AB, AC respectively, such that their middles are on the line HD. All these segments are tangent to the above parabolla.

MedialLine.html

Miquel_Point.html

Olympiad1.html

ParabolaProperty.html

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