About two years ago, I came across this wonderful property of conic sections while using Geogibra. I am not sure whether it is a new discovery or if it was previously discovered. If it was previously discovered, put a source mentioning it, and in any case, how can we prove that?
If we have a moving circle that touches a conic section at two points and its tangents are perpendicular to the line of the motion path and we draw the tangents of the segments at the points of their intersection with it, then the path of movement of the two points of intersection of the two tangents located outside the straight line of the motion path will make a conic section of the same type and with the same ratio between the large diameter and the distance between the focus and the center The two main vertices of the first conic section are considered as its focus.
In the case of an ellipse you can see the following link
$OF/OA=OA/OB$
In the case of an parabola you can see the following link
$FQ=QT=TV$
There are two cases of hyperbola:
$OF/OA=OA/OH$
amendment: Here is a link to a file in Arabic with some additional details on the subject that might be useful
https://drive.google.com/file/d/1YxStcytgtbCc7YlVKEF0ZCBdIhB9p6Cu/view?usp=drivesdk