## On Artzt parabolas of isosceli

There are various ways to generate the Artzt parabolas (see Artzt.html and references there). Here I discuss one particular to the isosceles and the parabola tangent to its equal legs.
Start with the parabola c itself and consider a point P varying on it. Take also two arbitrary points {A,C} on the symmetry axis M of the isosceles. Draw two lines through P: (i) line AP and (ii) the parallel MP to M from P. Second line intersects the basis L of the triangle at a point E. Define point Q to be the intersection point of lines AP and CE. As P moves on c point Q describes a conic c'.

The proof follows as a special case of a conic generation discussed in MaclaurinLike.html . The assumptions of the property proved there apply if one complements the two points {A,C} on line M with its point at infinity B. Then line MP defined above plays the role of BP in the aforementioned reference.
By the discussion in that reference also we know that there is a projectivity mapping the parabola c to the c', which is a perspectivity with homology ratio k = (A,O,B,C). This ratio is obtained also for all quadruples of points like (A,P',P,Q), where P' is the intersection point of AP with line L. In particular, when P goes to infinity (=B), then P' = O and Q=C, thus k = (A,O,B,C) = CO/CA.
Using this we can find the other intersection point C'of c' with the line M and construct the conic.

Problem
Find all positions of {A,C} for which the corresponding conic c' is a circle.
- As seen in ArtztIsosceles.html the circumcircle is such a case but there are infinite circles having this property.
- In fact every circle-member of the bundle through points {I, J} has this property.
- ArtztIsosceles3.html contains the relevant figure and statements.