Dear Alex,

I just finished calculation for general case, say "parallel with one cevians and cutting another cevians".

Our perpendicular case is one when "parallel with altitudes" and cutting another cevians (of incenter, circumcenter, orthocenter...).

The locus of all points such that we can make closed chain of six points is union of:
- One circumconic
- One circumcubic

When "parallel with altitudes" the circumconic is circumcircle and the circumcubic is Darboux cubic.

I will collect some special cases with simple points so may be we can find interesting elementar proof.

Best regards,
Bui Quang Tuan

>Have you tried drawing the lines which are not perpendicular
>but fall under the same inclination?