Quadrilateral

vertices, called and diagonal

QUADRILATERAL, in geometry, a figure formed by four straight lines. It is said to be plane or skew according as the four lines do or do not lie in one plane. Quad rilateral is also a military term applied to a combination of four fortresses mutually supporting each ,other. The most famous military quadrilateral was that of the four fortified towns of north Italy—Man tua and Peschiera on the Mincio, and Ve rona and Legnago on the Adige. This quadrilateral gave Austria a firm hold on Lombardy.

Let us consider the plane figure bounded by four lines termi nated at the vertices. A line joining a pair of opposite vertices is called a diagonal. The area of such a quadrilateral is half the product of the length of one diagonal by the sum of the perpen diculars drawn to this diagonal from the other two angular points. The sum of the squares of the four sides of such a quadrilateral is equal to the sum of the squares of its diagonals increased by four times the square of the line joining the midpoints of the diagonals. If the vertices of such a quadrilateral lie on a circle,

the product of the diagonals is equal to the sum of the products of the opposite sides.

It is in projective geometry that the quadrilateral plays its most interesting role. If A, B, C, D are four points in a plane no three of which are collinear, then the lines AB, BC, CD, DA, each taken to be of indefinite extent, form a quadrilateral. These sides intersect in pairs not only in the four given points (called vertices) but also in a point E on both AB and CD, and in a point F on both BC and AD. The points E and F are also called vertices. Then each side of the quadrilateral contains three ver tices. Two vertices not on the same side are called opposite. A. line joining two opposite vertices is called a diagonal. The config uration so described is called a complete quadrilateral. The dual figure is called a complete quadrangle. These figures are of funda mental importance in projective geometry (q.v.).