[alogo] 1. Artzt parabola in isosceles triangle

Consider an isosceles triangle ABC and its circumcircle O(r). Draw line a' parallel to its basis a to the same side with point A and at distance r from a. Take point P varying on the basis a and let P' be the intersection point with a' of the line orthogonal to a and passing through P.
[1] Draw OP' and the chord QQ' through P and parallel to OP'. Q' is a fixed point on the circle.
[2] OAP'P and OP'PQ' are parallelograms. Quadrangle OPQP' is an isosceles trapezium.
[3] The intersection point W of the diagonals of the trapezium describes a parabola c.
[4] This parabola is the Artzt parabola passing through {B,C} and tangent there to the sides of the triangle.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]

The proofs are easy.
[1] follows from the fact that drawing OQ' orthogonal to a, the resulting OP'PQ' is a parallelogram.
[2] follows from [1] and the equality OQ = OA = P'P = r.
[3] follows from the fact that the medial line tW of OP' passes through W, which coincides with the intersection point of this line tW and the vertical line P'W to a' at P'. This is a standard definition of the parabola. W describes a parabola with focus at O and directrix the line a'.
[4] follows from the fact that tW obtains the positions of {AB,AC} when OP' is respectively orthogonal to these two lines (see Artzt.html for the definition of these parabolas).

[alogo] 2. Homography mapping circle to parabola

As is well known (see Projectivity.html ) a projectivity is defined by prescribing the image points {Yi, i=1,..4} of four points {Xi, i=1,...,4} in general position. Thus there is a unique projectivity F with the properties:
(i) F fixes points {B,C, O} and
(ii) F maps A to W0.
- It follows easily (exercise) that F fixes every point of line BC and also leaves invariant every line passing through O (maps such a line into itself fixing O and its intersection point with BC).
- Using this and the invariance of line AO one sees easily that every line parallel to BC maps to a line also parallel to BC. In particular the parallel to BC through A maps to the parallel to BC through W0.
- Using cross ratios allong line AO one sees easily that the tangent line t of the circle at Q' maps via F to the line at infinity.
- From the previous result follows also immediately that the tangents {tB, tC} to the circle, map via F to sides {BA, CA} respectively.
- From these facts follows, that points {A,B,C} and the tangents to the circle at these points map to {W0,B,C} and the tangents to the parabola at these points correspondingly. Thus, also the circle maps via F to the Artzt parabola c.

[alogo] 3. Artzt parabola and circumcircle

The previous properties indicate a relation between the Artzt parabola and the circumcircle of the triangle. This relation does not generalize immediately to arbitrary triangles, their circumcircles and their Artzt parabolas. There is though a generalization involving the Steiner outer ellipse. The discussion related to these matters continues to ArtztSteiner.html .

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