Lemoine hexagon Lemoine hexagon In geometry, the Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. There are two definitions of the hexagon that differ based on the order in which the vertices are connected. The Lemoine hexagon can be drawn defined in two ways, first as a simple hexagon with vertices at the intersections as defined before. The second is a self-intersecting hexagon with the lines going through the symmedian point as three of the
Sum of angles of a triangle Sum of angles of a triangle In several geometries, a triangle has three "vertices" and three "sides", where three angles of a triangle are formed at each vertex by a pair of adjacent sides. In a Euclidean space, the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180 °, radians, two right angles, or a half-turn. It was unknown for a long time whether other geometries exist, where this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century. Ultimately, the