[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 BB

[4] The middles D

[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 B

[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 B

[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 BB

[4] The middles D

[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 B

[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 B

[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 ABB

CoordinatesTransform.html

CoordinatesTransform2.html

Parabola.html

ParabolaProperty.html

ParabolaSkew.html

ThalesParabola.html

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