Figure 12 - uploaded by Dan Reznik
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Lines connecting each cusp P i to the center P i of the circle which osculates E at the pre-image Pi concur at M . Note these lines are diameters of said circles. Therefore M is the perspector of T and T , i.e., the ratio of their areas is 4. This perspectivity implies C2, X 2 , M are collinear. Surprisingly, the X 2 coincides with the circumcenter X3 of the pre-image triangle T (not drawn).
Source publication
The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties.
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We revisit constructions based on triads of conics with foci at pairs of vertices of a reference triangle. We find that their 6 vertices lie on well-known conics, whose type we analyze. We give conditions for these to be circles and/or degenerate. In the latter case, we study the locus of their center.
