In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point. They are in contrast to parallel lines.
In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors:
A triangle's altitudes run from each vertex and meet the opposite side at a right angle. The point where the three altitudes meet is the orthocenter.
Angle bisectors are rays running from each vertex of the triangle and bisecting the associated angle. They all meet at the incenter.
Medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid.
Perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter.
Other sets of lines associated with a triangle are concurrent as well. For example:
Any median (which is necessarily a bisector of the triangle's area) is concurrent with two other area bisectors each of which is parallel to a side.
A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle.
A splitter of a triangle is a line segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the Nagel point of the triangle.
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter, and each triangle has one, two, or three of these lines. Thus if there are three of them, they concur at the incenter.
The Tarry point of a triangle is the point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangle's first Brocard triangle.
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In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter s.
En géométrie affine, le milieu d'un segment est l'isobarycentre des deux extrémités du segment. Dans le cadre plus spécifique de la géométrie euclidienne, c'est aussi le point de ce segment situé à égale distance de ses extrémités. Symétrie centrale Deux points distincts A et A sont symétriques par rapport à un point O si et seulement si O est le milieu du segment [AA]. Dans la symétrie centrale de centre O, le symétrique de O est O lui-même. L'ensemble des points du plan équidistants de deux points A et B constitue la médiatrice du segment [AB].
vignette|Sur cette figure, les points a1,a2,a3 sont alignés, ainsi que les points b1,b2,b3. En revanche, les points a1,a2,b3 ne sont pas alignés. En géométrie, l’alignement est une propriété satisfaite par certains familles de points, lorsque ces derniers appartiennent collectivement à une même droite. Deux points étant toujours alignés en vertu du premier axiome d’Euclide, la notion d’alignement ne présente d’intérêt qu’à partir d’une collection de trois points.
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