MedialParabola

Parabola tangent to the sides and one medial line

Construct the parabola tangent to the three sides of a triangle ABC and the medial line of its side BC. The construction of the parabola is based on the property of all of its tangents, at its points P, to define segments bisected by the medial line HD. In fact, since BC will be tangent to the parabola and bisected by HD, which will be also tangent to the parabola, the same will be valid for every tangent of the parabola (look at ParabolaProperty.html ).

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.

Parabola tangent to the sides and one medial line

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