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In Geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral; it passes through several important points determined from the triangle. It passes through the orthocenter, the circumcenter, the centroid, and the center of the nine-point circle of the triangle.
Euler (1767) showed that in any triangle, the orthocenter, circumcenter, centroid, and nine-point center are collinear. In equilateral triangles, these four points coincide, but in any other triangle they do not, and the Euler line is determined by any two of them. The center of the nine-point circle lies midway along the Euler line between the orthocenter and the circumcenter, and the distance from the centroid to the circumcenter is half that from the centroid to the orthocenter.
Here are some triangle centers that lie on the Euler line:
X(2), X(3), X(4), X(5), X(20), X(21), X(22), X(23), X(24), X(25), X(26), X(27), X(28), X(29), X(30), X(140), X(186), X(199), X(235), X(237), X(297), X(376), X(377), X(378), X(379), X(381), X(382), X(383), X(384), X(401), X(402), X(403), X(404), X(405), X(406), X(407), X(408), X(409), X(410), X(411), X(412), X(413), X(414), X(415), X(416), X(417), X(418), X(419), X(420), X(421), X(422), X(423), X(424), X(425), X(426), X(427), X(428), X(429), X(430), X(431), X(432), X(433), X(434), X(435), X(436), X(437), X(438), X(439), X(440), X(441), X(442), X(443), X(444), X(445), X(446), X(447), X(448), X(449), X(450), X(451), X(452), X(453), X(454), X(455), X(456), X(457), X(458), X(459), X(460), X(461), X(462), X(463), X(464), X(465), X(466), X(467), X(468), X(469), X(470), X(471), X(472), X(473), X(474), X(475), X(546), X(547), X(548), X(549), X(550), X(631), X(632).