CONCURRENCE (ML. concurrentia, con currence, from concurrcre, to run together, from con-, together + currcrc. to run) AND COL LINEARITY (from Lat. corn-, together + linca, line). If several lines have a point in common they arc said to be concurrent. The common point is called the focus or vertex of the pencil of lines. if several points lie on one straight line they are said to be collinear. The line is called the base of the range of points. That portion of geometry concerned with concurrent lines and collinear points is called the theory of concur rence and collinearity. Some of its fundamental propogitions are: If a transversal intersects the sides of a tri angle ABC in the points X, Y, Z, the segments of the sides of the triangle are connected by the relation (AZ:ZB) • (BX:XC) • (CY:YA)= — 1. Conversely, if the points be so taken that the rela tion holds, then the three points arc collinear. (This relation is known as Menelaus's theorem.) lf the three lines AO, BO. CO drawn from the vertices of the triangle ABC are concurrent in 0 and meet the opposite sides in X. I', Z. then BX • CY'AZ =-- —CX•AY•BZ, and conversely (Ceva's theorem).
If three lines perpendicular to the sides of a triangle ABC at X. Y. Z are concurrent. then -h = 0.
Conversely, if this relation holds, the per pendiculars are concurrent.
If the lines joining the vertices of two tri angles are concurrent, their corresponding sides intersect in three collinear points. (This proposi tion, known as Desargues's theorem, is true for any rectilinear figures.) The opposite pairs of sides of a hexagon in scribed in a conic intersect in three collinear points (Pascal's theorem).
The lines joining the opposite vertices of a hexagon circumscribed about a conic are con current (Brianchon's theorem).
The polars of a range of points with respect to a circle (q.v.) are concurrent. and conversely.
If from any point on a circle perpendiculars are drawn to the sides of an inscribed triangle. their feet are collinear. (The base of this range is canes' Simson's line.) From these and other similar theorems, many properties of elementary geometry follow at owe; as, the altitudes of a triangle are concurrent, the medians of a triangle arc concurrent, etc. The theorems of Pascal and Brianclum lead to numer ous theorems in modern geometry. Consult: Cre mona, Elements of Projeetire Geometry. trans. by Leudesdorf (Oxford, 1885) : Casey, Segue/ to Euclid (Dublin, 1SSS); Beman and Smith, New Plane and Solid Geometry ( Boston, 1900) ; Geometry of the Circle (New York, 1891).