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Constructing these hyperbolae

I have an unusual way of constructing the Feuerbach point F_o that sheds light on this conic (as well as leading to a stable construction of this point in Sketchpad). I construct F_o using the affine parallelogram of the conjugates G_o and N_o. The G_o—N_o line is one diagonal, the G—F_o line is the other. Not knowing where F_o is, we find the intersection of this 2nd diagonal with the circumcircle. This point is dF_o. Take the medial of this point and you have F_o. But better still reflect dF_o though O to get ogF_o (where the prefix "o" stands for reflection through the circumcenter). This is the fourth intersection of the Feuerbach conic with the circumcircle and is thus another point on this curve. This construction is part of my general affine theory and constructs the center of many circumconics.

Constructing the Feuerbach Point and the original Feuerbach hyperbola

Begin with triangle ABC, its centroid and its circumcircle.

Construct the Gergonne and Nagel points along with the segment that joins them. This segment is the Mineur line and will be a diagonal of the affine parallelogram.
The isotomic of this line is the Feuerbach hyperbola.

Construct the duals (or the tripolars) of G_o and N_o, which concur at the perspector of the the Feuerbach hyperbola.

Construct the medial of the Nagel point (the incenter) and the medial of the Gergonne point (the Mittenpunkt). Connect these to N_o and G_o as shown, constructing Q, a parallelogram vertex.

The affine parallelogram is completed by creating the point dQ (the dilated or antimedial of Q).

The second diagonal is the line G-conic center. Find its intersection with the circumcircle (the one on the side of dQ). The medial of this point is F_o, the center of the Feuerbach hyperbola.

There are now four ways to construct the conic: (1) we know the perspector. (2) we know the center. (3) We know 7 points on it: ABC, I_o, N_o, G_o, Mo; and (4) it is the isotomic of G_o-N_o.
The fourth intersection is odF_o the reflection of dF_o through O.

The axes are constructed as Simson lines of the intersections of O-I_o or from the intersections of G_o-N_o with the Steiner ellipse. The axes are the bisectors of the asymptotes.

Constructing the Feuerbach Point and the original Feuerbach hyperbola
	Begin with triangle ABC, its centroid and its circumcircle.
	Construct the Gergonne and Nagel points along with the segment that joins them. This segment is the Mineur line and will be a diagonal of the affine parallelogram. The isotomic of this line is the Feuerbach hyperbola.
	Construct the duals (or the tripolars) of G_o and N_o, which concur at the perspector of the the Feuerbach hyperbola.
	Construct the medial of the Nagel point (the incenter) and the medial of the Gergonne point (the Mittenpunkt). Connect these to N_o and G_o as shown, constructing Q, a parallelogram vertex.
	The affine parallelogram is completed by creating the point dQ (the dilated or antimedial of Q).
	The second diagonal is the line G-conic center. Find its intersection with the circumcircle (the one on the side of dQ). The medial of this point is F_o, the center of the Feuerbach hyperbola.
	There are now four ways to construct the conic: (1) we know the perspector. (2) we know the center. (3) We know 7 points on it: ABC, I_o, N_o, G_o, Mo; and (4) it is the isotomic of G_o-N_o. The fourth intersection is odF_o the reflection of dF_o through O.
	The axes are constructed as Simson lines of the intersections of O-I_o or from the intersections of G_o-N_o with the Steiner ellipse. The axes are the bisectors of the asymptotes.