ArtztIsosceles2

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 M_P 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 M_P 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.

On Artzt parabolas of isosceli

See Also