is the incircle of the medial triangle, and its center is the center of mass of the perimeter of the triangle.) Any vertex, the tangency of the opposite side with the incircle, and the Gergonne point are collinear. From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle. The lines connecting the feet of the altitudes intersect the opposite sides at collinear points. A triangle's incenter, the midpoint of an altitude, and the point of contact of the corresponding side with the excircle relative to that side are collinear. Menelaus' theorem states that three points on the sides (some extended) of a triangle opposite vertices respectively are collinear if and only if the following products of segment lengths are equal: The incenter, the centroid, and the Spieker circle's center are collinear. The circumcenter, the Brocard midpoint, and the Lemoine point of a triangle are collinear. Two perpendicular lines intersecting at the orthocenter of a triangle each intersect each of the triangle's extended sides. The midpoints on the three sides of these points of intersection are collinear in the Droz–Farny line. Quadrilaterals In a convex quadrilateral whose opposite sides intersect at and , the midpoints of are collinear and the line through them is called the Newton line (sometimes known as the Newton-Gauss line). If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line. In a convex quadrilateral, the quasiorthocenter , the "area centroid" , and the quasicircumcenter are collinear in this order, and . (See Quadrilateral#Remarkable points and lines in a convex quadrilateral.) Other collinearities of a tangential quadrilateral are given in Tangential quadrilateral#Collinear points. In a cyclic quadrilateral, the circumcenter, the vertex centroid (the intersection of the two bimedians), and the anticenter are collinear. In a cyclic quadrilateral, the area centroid, the vertex centroid, and the intersection of the diagonals are collinear. In a tangential trapezoid, the tangencies of the incircle with the two bases are collinear with the incenter. In a tangential trapezoid, the midpoints of the legs are collinear with the incenter. Hexagons Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order