## Παρασκευή, 3 Ιουνίου 2016

### MAPPINGS OF THE EULER LINE ONTO ITSELF

NPC centers

1. Let ABC be a triangle and N1, N2, N3 = the NPC centers of OBC, OCA, OAB, resp.

The NPC center of N1N2N3 lies on the Euler line of ABC

Let P be a point on the Euler line of ABC and M1, M2, M3 the midpoints of PN1, PN2, PN3, resp.

The NPC center of M1M2M3 lies on the Euler line of ABC. (ie it is the intersection of the Euer lines of ABC and M1M2M3).

2. Let ABC be a triangle and Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.

The circumcenter of NaNbNc lies on the Euler line of ABC

Let P be a point on the Euler line of ABC and Ma, Mb, Mc the midpoints of PNa, PNb, PNc, resp.

The O (circumcenter) of MaMbMc lies on the Euler line of ABC. (ie it is the intersection of the Euler lines of ABC and MaMbMc.)

3. Let ABC be a triangle and Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

The circumcenter of NaNbNc lies on the Euler line of ABC

Let P be a point on the Euler line of ABC and Ma, Mb, Mc the midpoints of PNa, PNb, PNc, resp.

The O (circumcenter) of MaMbMc lies on the Euler line of ABC. (ie it is the intersection of the Euler lines of ABC and MaMbMc).

Which other (than N,I) points have that property?

4. Let ABC be a triangle and Na, Nb, Nc = the NPC centers of GBC, GCA, GAB, resp.

The centroid of NaNbNc lies on the Euler line of ABC

Let P be a point on the Euler line of ABC and Ma, Mb, Mc the midpoints of PNa, PNb, PNc, resp.

The G (centroid) of MaMbMc lies on the Euler line of ABC. (ie it is the intersection of the Euler lines of ABC and MaMbMc).

Circumcenters

1. Let ABC be a triangle and Oa, Ob, Oc = the circumcenters of IBC, ICA, IAB, resp.

The circumcenter of OaObOc lies on the Euler line of ABC

Let P be a point on the Euler line of ABC and Ma, Mb, Mc the midpoints of PNa, PNb, PNc, resp.

The O (circumcenter) of MaMbMc lies on the Euler line of ABC. (ie it is the intersection of the Euler lines of ABC and MaMbMc).

2. Let ABC be a triangle and Oa, Ob, Oc = the circumcenters of NBC, NCA, NAB, resp.

The NPC center of OaObOc lies on the Euler line of ABC

Let P be a point on the Euler line of ABC and Ma, Mb, Mc the midpoints of PNa, PNb, PNc, resp.

The NPC center of MaMbMc lies on the Euler line of ABC. (ie it is the intersection of the Euler lines of ABC and MaMbMc).

3. Let ABC be a triangle and Oa, Ob, Oc = the circumcenters of GBC, GCA, GAB, resp.

The centroid of OaObOc lies on the Euler line of ABC

Let P be a point on the Euler line of ABC and Ma, Mb, Mc the midpoints of PNa, PNb, PNc, resp.

The G (centroid) of MaMbMc lies on the Euler line of ABC. (ie it is the intersection of the Euler lines of ABC and MaMbMc).

Antreas P. Hatzipolakis, 3 June 2016