Quadrilateral

A rectangle ( also quadrangle outdated, even Tetragon ) is a figure of plane geometry, namely a polygon with four corners and four sides. In mathematics (hence ) is defined ( flat ) quadrangles as polygons with four corners and four edges ( or sides ). Analogous to triangles is also possible a generalization of the square term on non-Euclidean geometries ( curved squares). In projective geometry complete quadrangles and its dual complete four page play an important role. In the finite geometry incidence properties of the quadrangle to the definition of " Generalized Quadrangle " are used.

The regular (or regular ) square is the square.

A quadrilateral has two diagonals. If both diagonals in the quadrilateral, the quadrilateral is convex ( convex quadrilateral ), exactly one diagonal outside, so the square has a concave corner ( non- convex quadrilateral ). In a battered about (also: crossed ) square both diagonals are outside of the quadrilateral (as an example see entangled trapezoid). About Beaten squares are generalized polygons and are not normally counted among the ( normal or "real" ) squares. The same applies to degenerate quadrilaterals in which two or more vertices coincide or more than two vertices lie on a straight line.

• The sum of the interior angles in a quadrilateral is 360 degrees or 2π.

Special quadrilaterals

A trapezoid is a quadrangle having at least two parallel sides. Are any two opposite sides are parallel, one speaks of the parallelogram. A quadrilateral having four equal-sized (interior) angles (90 °, see right angle) has, is a rectangle. When kite quadrilateral ( deltoid ) the diagonals are perpendicular, and one diagonal is bisected by the other. This is equivalent to saying that there are two pairs of adjacent sides, each of the same length. In four equal sides is called a rhombus (diamond ). A square has four equal sides and four equal (interior) angle ( 90 °). For a cyclic quadrilateral, the four sides of the perimeter tendons. Are the four sides of an inscribed circle tangents, so it is called a tangent quadrilateral.

Between the various types of quadrilateral set relations in particular the subset relationships shown in the graph, such as apply:

• ⊂ ⊂ squares rectangles parallelograms ⊂ ⊂ Trapeze convex quadrilaterals

( In each case, the term is synonymous with X the set of all X)

Furthermore, the following relations hold:

• Squares rectangles = ∩ diamonds
• Squares = dragon squares ∩ isosceles trapezoids
• Rectangles = tendons quadrilaterals parallelograms ∩
• Diamonds = dragon squares ∩ Trapeze
• Diamonds = tangent squares ∩ parallelograms
• Isosceles trapezoids = tendons squares ∩ Trapeze

Classification

The planar quadrilaterals are classified according to various criteria:

• On the properties of the interior: convex
• Not convex

• For symmetry properties: a diagonal symmetry axis: Deltoid (dragon square)
• Both diagonal axes of symmetry are: diamond (diamond )
• The perpendicular bisector of a side is an axis of symmetry: isosceles trapezoid
• The perpendicular bisectors of two sides are symmetry axes: rectangle
• Four axes of symmetry: Square
• Two-fold symmetry ( point symmetry ): parallelogram
• Fourfold symmetry: Square

• According to the length of the site: two pairs of opposite sides of equal length: parallelogram
• Equilateral quadrangle: diamond (diamond )
• The sum of the lengths of opposite sides is equal to: tangent rectangle

• The size of the angle: two pairs of equal size opposite angle: parallelogram
• Two pairs of equal-sized adjacent angle: isosceles trapezoid
• Equiangular quadrilateral: square
• The sum of opposite angle is 180 °: cyclic quadrilateral

• According to the position of Pages: a pair of parallel sides: trapeze
• Two pairs of parallel sides: parallelogram
• The sides touch the same circle ( the inner circle ): tangent quadrilateral

• According to the position of the corners: the vertices lie on a circle ( the radius): cyclic quadrilateral

Formulas

The rectangular area A can be calculated from determined

If the coordinates of the vertices given, we obtain the Gaussian trapezoidal formula the simple expression

A convex quadrilateral can be described by five independent determination as a combination of the following pieces of information:

• Angles at the corners ( interior angle )
• Length of the sides
• Length of the diagonal
• Scope
• Surface.

If you leave even non-convex quadrilaterals, so are some combinations, eg " 4 pages and 1 interior angle " however ambiguous, as the given angle opposite corner may be convex or concave.

Center of gravity

In point-symmetric quadrilaterals ( parallelograms ), the focus is the center of symmetry, ie, the diagonal intersection.

In general one has to distinguish between the corner of gravity ( all mass sits in the corners, each corner has the same mass) and the centroid ( the mass is evenly distributed over the area of ​​the quadrilateral. ) At the triangle these two priorities are consistent. There are also the edges of gravity ( the mass is distributed evenly to the edges, the mass of each edge is proportional to its length). The edge focus is rarely considered. He is wrong even when the triangle with the area and corner focus match, but there corresponds the incenter of the triangle center.

The centroid of a quadrilateral can be constructed as follows: You divide the square by a diagonal into two triangles and each determined their focus as the intersection of the medians. These two points are connected by a straight line. The same is repeated one by dividing the square by the other diagonal. The intersection of the two straight lines connecting the center of gravity of the quadrangle.

Reason: The straight line through the two triangular areas is a heavy line of both triangles and hence also of the quadrangle. So the focus must be on this line.

The corner focus is obtained by connecting the midpoints of opposite sides. The intersection of the two lines connecting the center of gravity area. If a Cartesian coordinate system is given, one can calculate the coordinates of the corner of the center of gravity from the coordinates of the corners: