ParabolaFromProjections

Parabola through projections

Given a triangle ABC, project a point D on the base BC to points E, F on sides AC, AB respectively. The following facts, related to some constructs with D varying on BC, are true:
[1] The circumcircles of triangles AFE pass through D and the foot P of the altitude from A of triangle ABC.
[2] Triangle PEF has fixed angles, thus PEF varies by remaining always similar to itself as D glides on BC.
[3] The two altitudes BB₀, CC₀ of ABC are particular positions of line EF taken when D coincides with B and C respectively.
[4] The middles D₀ of lines EF lie on a line L passing through the middles of the other altitudes BB₀, CC₀.
[5] Vertex D' of parallelogram AE'DF moves on a line L' parallel to L at double distance from A than that of L.
[6] Line EF envelopes a parabola coinciding with the Artzt parabola relative to side B'C' of triangle AB'C'.
[7] The focus of this parabola is P and the tangent at its vertex is line B₁C₁ joining the projections of P on sides AB, AC.
[8] Altitude AP of triangle ABC is the symmedian from A for triangle AB'C'.
[9] The circumcenter of triangle ABC is on the median AM of triangle AB'C'. The circumcenter O of triangle AB'C' is on the side BC of triangle ABC.
[10] The radical axis of the circumcircles of ABC and AB'C' passes through the intersection point G of lines BC and B₁C₁ (tangent at the vertex of the parabola).

[1] D is viewing EF under pi-A. P is viewing the diameter of this circle under a right angle.
[2] AFPDE is a pentagon inscribed in the circumcircle of AEF.
[3] The two altitudes BB₀, CC₀ of ABC are particular positions of line EF taken when D coincides with B and C respectively.
[4] The middles D₀ of lines EF lie on a line L passing through the middles of the other altitudes BB₀, CC₀.
[5] Vertex D' of parallelogram AE'DF moves on a line L' parallel to L at double distance from A than that of L.
[6] Line EF envelopes a parabola coinciding with the Artzt parabola relative to side B'C' of triangle AB'C'.
[7] The focus of this parabola is P and the tangent at its vertex is line B₁C₁ joining the projections of P on sides AB, AC.
[8] Altitude AP of triangle ABC is the symmedian from A for triangle AB'C'.
[9] The circumcenter of triangle ABC is on the medina AM of triangle AB'C'. The circumcenter O of triangle AB'C' is on the side BC of triangle ABC.
[10] The radical axis of the circumcircles of ABC and AB'C' passes through the intersection point G of lines BC and B₁C₁ (tangent at the vertex of the parabola).

Remarks
[1] The discussed properties show that the relation between lines BC (described in vertical coordinates) and B'C' (described in parallel coordinates) involves the ideas of altitudes and symmedians of the two triangles.
[2] Point D is also the orthocenter of the variable triangle D'EF, thus line DD' is always orthgonal to line EF.
[3] Line B'C' is easily constructible from ABC by noticing that ABB₂B₀ and ACC₂C₀ are the positions of parallelogram AED'F when D coincides with B and C respectively.

Parabola through projections

See Also