ORTHOPOLE CIRCLES

Let (a,b,c,d) be a complete quadrilateral.

Denote:

A(b,c,d), B(c,d,a),C(d,a,b), D(a,b,c) = the antimedial triangles of (b,c,d), (c,d,a),(d,a,b), (a,b,c), resp.

Oa, Ob, Oc, Od = the circumcenters of A(b,c,d), B(c,d,a),C(d,a,b), D(a,b,c), resp. [=orthocenters of (b,c,d), (c,d,a),(d,a,b), (a,b,c), lying on the Steiner line]

M = the Miquel point of the quadrilateral (= the point of concurrence of the circumcircles of (b,c,d), (c,d,a),(d,a,b), (a,b,c))

Are the orthopoles of MOa, MOb, Moc, MOd with respect A(b,c,d), B(c,d,a),C(d,a,b), D(a,b,c) concyclic?

The orthopoles lie on the NPCs of A(b,c,d), B(c,d,a),C(d,a,b), D(a,b,c) [ = circumcircles of (b,c,d), (c,d,a),(d,a,b), (a,b,c)]

Antreas P. Hatzipolakis, 19 July 2013

CONCURRENT CIRCLES - ORTHOCENTERS - ISOGONAL CONJUGATE POINTS

Let ABC be a triangle and P,P* two isogonal conjugate points. Denote: H1,H2,H3 = the orthocenters of PBC, PCA, PAB, resp. and Ha,Hb,Hc = the orthocenters of P*BC, P*CA, P*AB, resp.

The circumcircles of: (1) H1HbHc, H2HcHa, H3HaHb (2) HaH2H3, HbH3H1, HcH1H2 are concurrent.
Antreas P. Hatzipolakis, 5 July 2013