Let ABC be an equilateral triangle and P a point.
Which same central lines of the triangles PBC, PCA, PAB
are concurrent for all P's ? The OH (Euler) lines? The OI lines? ......
Antreas P. Hatzipolakis, Hyacinthos #21368
Central lines Ln (OH,OK,OI....) as described in the paper:
Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu: Concurrency of Four Euler Lines, Forum Geometricorum Volume 1 (2001) 59–68 [pdf file]
The Euler line of equilateral is any line passing through the center of the triangle.
So the Euler Lines of ABC, PBC, PCA (and also of ABC, PCA, PAB and of ABC, PAB, PBC) are concurrent on the Euler line of PBC and therefore the Euler lines of PBC, PCA, PAB are concurrent..