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X(1), X(25), X(69)

P (pivot), P* (its isogonal conjugate)

excenters

The OXI is an isogonal pK with pivot P = X(14826) = X(2)X(98)/\X(25)X(69), (Jean-Pierre Ehrmann Hyacinthos #8452).

P* is unlisted in ETC with SEARCH = –2.69701378845516.

K173 is described by Antreas Hatzipolakis as follows :

Let ABC be a triangle, P a point (not on the sidelines of ABC) and PaPbPc the pedal triangle of P.

The parallel La from P to BC intersects AB, AC at Lab, Lac, resp. The locus of P such that ABC, A'B'C' are perspective is the OXI cubic in the following configurations :

• Let Pab = orthogonal projection of Pa on AB, Pac = orthogonal projection of Pa on AC, Ab = PaPab /\ La, Ac = PaPac /\ La, A' = BAb /\ CAc. Similarly B', C'. (Hyacinthos #8448)
• The perpendicular to AB at Lab intersects PPa at Ab. The perpendicular to AC at Lac intersects PPa at Ac. A' = BAb /\ CAc. Similarly B', C'. (Hyacinthos #8462)
• Let Oab, Oac = the circumcenters of the triangles BPaLab, CPaLac, resp. A' = BOab /\ COac. Similarly B', C'. (Hyacinthos #8478)
• Let Hab, Hac = the orthocenters of the triangles BPaLab, CPaLac, resp. A' = BHab /\ CHac. Similarly B', C'. (APH, 30 October 2003)

Note: OXI (in Greek) = NO. The historical reply of Greece to fascist Italy on 28 October 1940.