Forum Geometricorum Volume 1 (2001) 59–68. FORUM GEOM ISSN 1534-1178 Concurrency of Four Euler Lines Antreas P.Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu Abstract. Using tripolar coordinates, we prove that if P is a point in the plane of triangle ABC such that the Euler lines of triangles PBC, AP C and ABP are concurrent, then their intersection lies on the Euler line of triangle ABC. The same is true for the Brocard axes and the lines joining the circumcenters to the respective incenters. We also prove that the locus of P for which the four Euler lines concur is the same as that for which the four Brocard axes concur. These results are extended to a family Ln of lines through the circumcenter. The locus of P for which the four Ln lines of ABC, PBC, AP C and ABP concur is always a curve through 15 finite real points, which we identify. 1. Four line concurrency Consider a triangle ABC with incenter I . It is well known [13] that the Euler lines of the triangles IBC , AIC and ABI concur at a point on the Euler line of ABC , the Schiffler point with homogeneous barycentric coordinates 1  a(s - a) b + c : b(s - b) c + a : c(s - c) a + b  . There are other notable points which we can substitute for the incenter, so that a similar statement can be proven relatively easily. Specifically, wehave the follow- ing interesting theorem. Theorem 1. Let P be a point in the plane of triangle ABC such that the Euler lines of the component triangles PBC , APC and ABP are concurrent. Then the point of concurrency also lies on the Euler line of triangle ABC . When one tries to prove this theorem with homogeneous coordinates, calcula- tions turn out to be rather tedious, as one of us has noted [14]. We present an easy analytic proof, making use of tripolar coordinates. The same method applies if we replace the Euler lines by the Brocard axes or the OI -lines joining the circumcen- ters to the corresponding incenters. Publication Date: April 9, 2001. Communicating Editor: Jean-Pierre Ehrmann. 1 This appears as X21 in Kimberling’s list [7]. In the expressions of the coordinates, s stands for the semiperimeter of the triangle.