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

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

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

Using this we can find the other intersection point C'of c' with the line M and construct the conic.

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.

ArtztIsosceles.html

ArtztIsosceles3.html

ArtztSteiner.html

MaclaurinLike.html

Perspectivity.html

Produced with EucliDraw© |