Let B and C be two fixed points on a circle with a center O
such that the points B, C, and O are not collinear. Let A be
a variable point on the same circle (distinct from points B
and C and the perpendicular bisector of BC). Let E and F
be the midpoints of BC and AO respectively. Let ray AE
intersect the circle again at point D. Let lines DO and EF
intersect at point P.

What is the locus of point P as point A moves around the
circle?
  

(In reply to re: GSP exploration by Charlie)

It looks like, when segment BC is closer to the center of the circle than 1/3 the radius, the constructed hyperbola encloses the original circle within one of its nappes. When chord BC is farther than 1/3 of the radius away from the center, one of the nappes intersects the circle in two points. At 1/3 radius distance from the center one nappe is tangent to the circle on the side closer to the other nappe.

Posted by Charlie on 2017-12-13 15:24:45