HYACINTHOS 23422

Antreas Hatzipolakis

 

Let ABC be a triangle.

Denote:

O1,O2, O3 = the circumcenters of OBC,PCA,OAB, resp

O12, O13 = the orthogonal projections of O1 on AC, AB, resp.

O23, O21 = the orthogonal projections of O2 on BA, BC, resp.

O31, O32 = the orthogonal projections of O3 on CB, CA, resp.

M1, M2, M3 = the midpoints of O12O13, O23O21, O31O32, resp.

Oa, Ob, Oc = the circumcenters of HBC, HCA, HAB, resp.

(= the reflections of H in BC,CA,AB, resp.)

Oab, Oac = the orthogonal projections of Oa on AC, AB, resp.
Obc, Oba = the orthogonal projections of Ob on BA, BC, resp.
Oca, Ocb = the orthogonal projections of Oc on CB, CA, resp.

Ma, Mb, Mc = the midpoints of OabOac, ObcOba, OcaOcb, resp.

1. The perpendicular bisectors of OabOac, ObcOba, OcaOcb are

concurrent at N

2. The triangles M1M2M3 and Orthic of ABC are perspective (homothetic).

Perspector (on the Euler line)?

 

3. The triangles M1M2M3, MaMbMc are perspective. Perspector?

4, The parallels through A,B,C to M1Ma, M2Mb, M3Mc, resp. are

concurrent. Point?

Generalizations (for two isogonal conjugate points P,P*)?

APH